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SI 



SELECTED PROPOSITIONS 



IN 



GEOMETRICAL CONSTRUCTIONS 



AND 



APPLICATIONS OF ALGEBRA TO GEOMETRY. 



BEING 



A KEY 



TO THE APPENDIX OF 



» 



DAVIES' LEGENDRE. 




A u° 

K- 5' -tcv?v 

A. S. BARNES & COMPANY, 

NEW YORK, CHICAGO and NEW ORLEANS. 

1875. 



9 A sa<? 

, L ^ 5 

1 S 75 

Davies* Course of Mathematics. 



I2ST THR.EE PARTS. 



I. COMMON SCHOOL COURSE. 

Davees' Primary Arithmetic — The fundamental principles displayed in Object 
Lessons. 

Davies' Intellectual Arithmetic— Referring all operations to the unit 1 as the only- 
tangible basis for logical development. 

Davies' Elements of Written Arithmetic— A practical introduction to the whole 
subject. Theory subordinated to Practice. 

Davies' Practical Arithmetic— A combination of Theory and Practice, clear, exact, 
brief, and comprehensive. 

II. ACADEMIC COURSE. 

Davies' University Arithmetic— Treating the subject exhaustively as a science, in a 
logical series of connected propositions. 

Davies' Elementary Algebra. — A connecting link, conducting the pupil easily from 
arithmetical processes to abstract analysis. 

Davies' University Algebra. — For institutions desiring a more complete but not the 
fullest course in pure Algebra. 

Davies' Practical Mathematics. — The science practically applied to the useful arts, 
as Drawing, Architecture, Surveying, Mechanics, etc. 

Davies 1 Elementary Geometry.— The important principles in simple form, but with 
all the exactness of rigorous reasoning. 

Davies' Elements of Surveying.— Re-written in 1870. A simple and full presentation 
for Instruction and Practice. 

III. COLLEGIATE COURSE. 

Davtes' Bourdon's Algebra.— Embracing Stui-m's Theorem, and a most exhaustive 
course. Re-written, in 1873. 

Davies' University Algebra. — A shorter course than Bourdon, for Institutions hav- 
ing less time to give the subject. 

Davies' Legendre's Geometry.— A standard work in this country and in Europe. 

Davies' Analytical Geometry.— A full course of Analysis, embracing the applications 
to surfaces of the second order. 

Davies' Differential and Integral Calculus, on the basis of Continuous Quantity 
and Consecutive Differences. 

Davies' Analytical Geometry and Calculus.— The shorter treatises, combined in 
one volume. 

Davies' Descriptive Geometry.— With application to Spherical Trigonometry, Spher- 
ical Projections, and Warped Surfaces. 

Davies' Shades, Shadows, and Perspective. — A succinct exposition of the mathe- 
matical principles involved. 

Davies' Nature and Utility of Mathematics, Logically considered. 

Davies and Peck's Mathematical Dictionary, or Cyclopedia of Mathematics. 



eA 



Entered according to Act of Congress, in the year 1875, by 
CHARLES DAVIES, <^ 
Da the Office of the Librarian of Congress, at Washington. 



PREFACE. 



The applications of Mathematical Science, to the Mechanic Arts, 
have received special attention within the past few years ; and no 
system of instruction, in any department, is now complete, unless it 
extends beyond the theoretical and into the field of Practical 
Knowledge. 

Geometry, in its essence and structure, is more purely abstract 
than any other branch of science. In its logical structure, it is 
conversant only about space; and yet, the relations and principles 
which it develops, afford the only basis of the true Practical. 

To extend Geometry to some of its most interesting and useful 
applications, an Appendix has been prepared and added to Legendre, 
embracing many Problems of Geometrical construction, and many 
applications of Algebra to Geometry : the whole being designed to 
explain and illustrate the methods of making Geometry a practical as 
well as a theoretical science. 

It would be unjust to those giving instruction, to add to their 
daily labors, the additional one, of finding appropriate solutions to so 



IV PREFACE. 

many difficult problems : hence, a Key has been made for their 
special use, in which the best methods of construction and solution 
are fully given. 

It is confidently hoped that this addition to a work which has 
been an accepted Text Book, both in this country and in Europe, for 
many years, may add something to its great value ; and it would be 
pleasant to indulge the hope that it will be received with a portion of 
that great favor which has been extended to the original work. 

FlSHKILL-ON-HuDSON, 

March, 1875. 



KEY. 




Proposition I. — Show that the bisectrices of two adjacent angles 
are perpendicular to each other. 

Demonstration. — Let DCB and DCA be two adjacent angles, 
and CP and CQ be their bisectrices. 

The angle PCD is equal to \ DCB by 
definition, and DCQ is equal to \ DCA : 
hence, PCD + DCQ = \ (DCB + DCA). 
The first member of this equation is equal 
to the angle QCP, and the second member 

is equal to a right angle (Bk. I. Prop. 1) ; hence, the angle QCP is 
a right angle, which was to be proved. 

Prop. II. — Show that the perimeter of any triangle is greater than 
the sum of the distances from any point within the triangle to its three 
vertices, and less than twice that sum. 

Demonstration. — Let ABC be any tri- 
angle, P any point within it, and PA, PB, 
PC, lines from P to the vertices. 

In accordance with Bk. I. Prop. 8, we have 
the following inequalities : 

AB -f BC > PA + PC 
BC -f CA > PA + PB 
AB + AC > PB + PC. 




6 KEY. 

Adding the first and second members of these inequalities, separately* 
and dividing by 2, we have, 

AB + BC + CA > PA + PB + PC, 

which proves the first part of the proposition. 

In accordance with Bk. I, Prop. 7, we have the following inequal- 
ities : 

AB < PA + PB 

BC < PB + PC 

CA < PC + PA. 

Adding the members of these inequalities, separately, and then fac- 
toring, we have, 

AB + BC + CA < 2 (PA + PB + PC), 

which proves the second part of the proposition. 

Prop. III. — Show that the angle between the bisectrices of two 
consecutive angles of any quadrilateral is equal to one-half the sum 
of the other two angles. 

Demonstration. — Let ABDE be any quadrilateral, and let AC and 
BC be the bisectrices of the angles A and B. 

The sum of the 4 angles of the quad- 
rilateral is equal to 4 right angles (Bk. I. 
Prop. 26, Cor. 1) ; or, denoting the angles by 
the letters A, B, D, E, and 1 right angle 
by R, we have, 

A + B-fI>-fE = 4R. 

Dividing both members of this equation by 2, factoring and trans- 
posing, we have, 

i (D + E) = 2R-i (A + B) . . . (1). 




PROPOSITIONS FROM LEGENDRE. 7 

But CAB and CBA are, respectively, equal to £ A and \ B ; hence, 
(Bk. I. Prop. 25, Cor. 1) we have for the angle ACB, denoted by C, 

C = 2R - i (A + B) . . . (2). 

Hence, in accordance with axiom 1, we have, from (1) and (2), 

C = \ (E + D), 
which was to be proved. 

Prop. IY. Theorem. — Shoiu that any point in the bisectrix of 
an angle is equally distant from the sides of the angle. 

Demonstration. — Let ABC be any angle, BP its bisectrix, and 
P any point of BP. 

Draw PR and PQ respectively perpendicular 
to the sides B A and BC ; then will PR and PQ be 
the distances from P, to BA, and BC. In the 
triangles BPR and BPQ, the angles PBR and 
PBQ are equal, by hypothesis, and the angles PRB 
and PQB are equal, because they are both right angles ; hence, the 
angles BPR and BPQ are equal (Bk. I. Prop. 25, Cor. 2). The 
triangles BPR and BPQ, have the angles PBR and BPR respec- 
tively equal to the angles PBQ and BPQ, and the side BP com- 
mon ; hence, they are equal in all their parts (Bk. I. Prop. 6), that- 
is, PR = PQ, ivhich was to be proved. 

Conversely; If PQ=PR, the line PB is the bisectrix of the angle 
ABC. 

Corollary. — If a line, as APC is drawn perpendicular to the' 
bisectrix of an angle, the parts intercepted between it and the side* 
of the angle will be equal : for the triangles PQC and PRA are equal 
in all their parts , hence, PC equals PA. 




8 



KEY. 




Prop. V. — If two sides of a triangle are prolonged beyond the 
third side, show that the bisectrices of the included angle and of the 
exterior angles, all meet in the same point. 

Demonstration. — Let ABC be any triangle, and let CO and BO 
be the bisectrices of the exterior angles 
ECB and DBC. 

Draw AO, and also, draw OE perpen- 
dicular to AE, OD perpendicular to AD, 
and OQ perpendicular to BC. It is to be 
shown that AO is the bisectrix of the 
ande EAD. 

Because CO is the bisectrix of ECB, we have, OE = OQ, and 
because BO is the bisectrix of DBC, OD = OQ; consequently, (Ax. 1) 
OD =p OE. Hence, from the last proposition, AO is the bisectrix of 
EAD, ivhich was to be proved. 

Prop. VI. — Show that the projection of a line on a parallel line 
is equal to the line itself ; and that the projection of a line on a line 
to ivhich it is oblique, is less than the line itself. 

Demonstration. — Let PQ be the projection of the parallel line 
AB, and also the projection of the oblique 
line AC. 

Because AB is parallel to PQ, and be- 
cause AP and BQ are both perpendicular to 
PQ and consequently parallel, the figure 
ABPQ is a parallelogram ; hence, AB = PQ, (Bk. I. Prop. 30), 
which proves the first part of the proposition. 

Because AB is perpendicular to QC, and AC is oblique to it, AC 
is greater' than AB, (Bk. I. Prop. 15) ; hence AC is greater than PQ, 
which proves the second part of the proposition. 




PKOPOSITIOJ^S FKOM LEGEKDEE. 




Prop. VII. — If a line is drawn through the point of intersection 
of the diagonals of a 'parallelogram, show that the line is bisected at 
the point. 

Demonstration. — Let ABCD be a parallelogram, AC and BD 
its diagonals, and their point of inter- p 

section. 

Draw the line PQ, through O, and 
limited by BC and AD. 

In the triangles POC and QOA, the 
sides AO and OC are equal (Bk. I. Prop. 31), the angles OOP and 
OAQ are equal (Bk. I. Prop. 20, Cor. 2), and the angles POC and 
QOA are equal, because they are opposite or vertical ; hence, the 
triangles are equal in all their parts (Bk. I. Prop. 6) : consequently, 
OP = OQ, which was to be proved. 

Prop. VIII. — The bisectrices of the four angles of any parallelo- 
gram form, by their intersection, a rectangle whose diagonals are 
parallel to the sides of the given parallelogram. \ , 

Demonstration.— Let ABCD be any parallelogram, and let AS> 
BS, CR, and DR be the bisectrices of its 
angles. 

Draw the diagonals PQ, RS, and 
prolong DA and BC to meet the bisec- 
trices CR and AS, in L and T. 

Because AS and BS are the bisec- 
trices of the angles A and B, their 
included angle is equal to one-half the sum of the angles D and 
(Prop. 3, Key); but the sum of D and C is two right angles; hence, 
the angle PSQ is a right angle. In like manner it may be shown 




10 KEY. 

that each of the angles SQR, QRP and EPS is a right angle ; hence, 
PRQS is a rectangle. 

Because DE and BK are parallels included between parallels, they 
are equal to each other (Bk. I. Prop. 28, Cor. 1). Since AP bisects the 
angle DAE and is perpendicular to DE, it divides the latter line into 
two equal parts, that is, EP is one-half of ED ; in like manner it may 
be shown that BQ is one-half of BK ; consequently EP is equal to 
BQ; hence, PQ is parallel to EB (Bk. I. Prop. 30). In the same 
way it may be shown that RS is parallel to LD, which was to be 
proved. 

Prop. IX. Theorem. — Show that the sum of the distances from 
any point in the base of an isosceles triangle to the other two sides, is 
equal to the distance from the vertex of either angle at the base, to the 
opposite side. 

Demonstration. — Let ABC be an isosceles triangle and let P be 
any point in its base. 

Draw CS parallel to AB, CD and QP perpen- 
dicular to AB, PR perpendicular to AC, and 
prolong QP to S. 

The angle PCS is equal to ABC, because they 
are alternate angles, and the angle PCR is equal to 
ABC by hypothesis; hence, the angles PCS and 
PCR are equal. In the triangles PCS and PCR, the angles PCS and 
PCR are equal, as just shown, the angles PRC and PSC are equal, 
because both are right angles, and consequently the remaining angles 
CPR and CPS are equal: these triangles also have the side CP 
common ; hence, they are equal in all their parts (Bk. I. Prop. 6) ; 
PR is therefore equal to PS, and QP + PR is equal to QP + PS, or 




PEOPOSITIOlSrS FKOM LEGENDKE. 



11 




to QS; but QS is equal to DC (Bk. I. Prop. 28, Cor. 1); hence, 
PQ _l. PR = CD, which was to be proved. 

Prop. X. — Show that the middle point of the hypothenuse of 
a right-angled triangle is equally distant from the three vertices. 

Demonstration. — Let ABC be a right-angled triangle, and let O 
be the middle point of its hypothenuse. 

Draw OB; also draw OP parallel to AB, 
and OQ parallel to CB. 

In the triangles OQA and CPO, the angles 
COP and OAQ are equal (Bk. I. Prop. 19, 
Sch.) ; the angles OCP and AOQ are equal 
for the same reason, and the sides CO and 
OA are equal by hypothesis ; hence, OP is equal to QA ; but OP 
is also equal to BQ, and consequently BQ is equal to QA. — In the 

triangles OQA and OBQ, the side OQ is common, BQ is equal to 
QA, and the included angles OQB and OQA are equal because both 
are right angles ; hence, OB is equal to OA, and to its equal to OC, 
which was to be proved. 

Prop. XI. — Draw two lines that shall divide a given right angle 
into three equal parts. 

Solution. — Let ADK be the given right angle. Take any dis- 
tance DL, and on it as a side, construct an 
equilateral triangle DLQ, and bisect the angle 
QDL by the line DS, (Bk. III. Probs. 10 and 5). 
The angle QDL is equal to two-thirds of a 
right angle (Bk. I. Prop. 25, Cor. 5) ; hence, 
KDS and its equal SDQ, are each equal to one- D 





12 KEY. 

third of a right angle and the remaining angle QDA is also equal to one- 
third of a right angle. The required lines are, therefore, DS and DQ. 

Prop. XII. — Draw a line AP through the vertex A of a triangle 
ABF and perpendicular to the bisectrix of the angle A ; construct a 
triangle PBF having its vertex P on AP and its base coinciding 
with that of the given triangle : then show that the perimeter of PBF 
is greater than that of ABF. 

Demonstration. — Let ABF be the given triangle, AG the bisec- 
trix of the angle A, AP perpendicular to AG, 
and let P be any point of AP. Prolong BA 
and make AM equal to AF ; draw MP. 

Because the angles BAF and FAM are 

adjacent, and AG is the bisectrix of the 

former, AP must be the bisectrix of the latter (Prop. 1, Key) ; hence 

the angles FAP and PAM are equal. In the triangles FAP and 

PAM, the side AP is common, AM is equal to AF by construction, 

and the included angles FAP and PAM are equal ; hence, PM is 

equal to PF. — In accordance with Bk. I. Prop. 7, we have the 

inequality : 

BP + PM > BA -f AM. 

Substituting PF for its equal PM, AF for its equal AM, and adding 
FB to each member, we have, 

BP + PF + FB > BA + AF + FB, 

which was to be proved. 

Prop. XIII. — Let an altitude of the triangle AB C be drawn from 
the vertex A, and also the bisectrix of the angle A ; then show that their 
included angle is equal to half the difference of the angles B and C. 



PBOPOSITICKN'S FEOM LEGE> T DEE. 



13 



Demonstration. — Let ABC be a triangle, AQ the bisectrix of 
the angle A, and AP the altitude drawn from 
the vertex A. The angle CAB is equal to 
two right angles diminished by the sum of the 
angles B and C ; hence, if we denote a right 
angle by R, we have, 

QAB = R 

In the right-angled triangle PAB, we have, 

PAB = R — B . . . (2). 

Subtracting (2) from (1), member from member, we have, 

QAB - PAB = £B - \ C, or QAP = \ (B - C), 

which vms to be proved. 



P-iC 




Prop. XIV. — Given two lines that would meet if sufficiently pro- 
longed, to draw the bisectrix of their included angle without finding 
its vertex. * 

Solution. — Let AB and CD be the given lines. 

Through any point P, of AB, draw PR 
parallel to CD (Bk. III. Prob. 6). Bisect the 
angle BPR by the line PQ, and draw the line 
PR perpendicular to PQ; bisect PR in S 
(Bk. III. Prob. 1), and draw SY parallel to 
PQ. 

Because CD is parallel to PR, the angle 
between the given lines is equal to the angle 

RPB, and consequently the required bisectrix is parallel to PQ; 
hence, PR is perpendicular to that bisectrix. Since S is the middle 




14 



KEY. 




point of PR, and SV is perpendicular to PR, it must be the required 
bisectrix (Prop. IV, Key, Cor.). 

Prop. XV. — From two points on the same side of a given line, 
draw two lines that shall meet each other at some point of the given 
line and make equal angles with that line. 

Solution. — -Let P and Q be the given points and AK the given 
line. 

Draw PIS' perpendicular to AK, and 
prolong it to M, making NM = NP ; 
draw MQ, and from the point R in which 
it intersects AK, draw RP. 

The right-angled triangles RNP and 
RNM, have the side PN equal to NM, by construction ; the side RN 
common and the included angles PNR and MNR are equal ; hence, 
the angle NRP is equal to the angle NRM; but the angle ARQ is 
cqualto its opposite angle NRM; hence, the lines PR and QR make 
equal angles with AK ; consequently they are the required lines. 

Prop. XVI. — Show that the sum of the lines drawn from two 
given points to any point of a given line, is the least possible when these 
lines are equally inclined to the given line. 

Demonstration. — Employing the same construction as in the last 
figure, let P and Q be the given points, 
and PR and QR equally inclined to AK; 
also PS and SQ unequally inclined. 

The line AK is drawn through the 
vertex R of the triangle QRP and perpen- 
dicular to the bisectrix of the angle QRP; 

hence, from Prop. XII, Key, QR -f RP < QS -|- SP, which was to be 
proved. 




PROPOSITIONS FEOM LEGENDRE. 



15 




B 



Prop. XVII. — From two given points, on the same side of a given 
line, draw two lines meeting on the given line, and equal to each other. 

Solution. — Let A and C be the given points and BD the given 
line. 

Draw AC, and bisect it by a perpendicular 
PQ ; from the point Q, in which this meets 
the given line, draw QA and QC. 

Because PQ is perpendicular to AC, at its 
middle point, Q is equidistant from A and C 
(Bk. I. Prop. 16) ; hence, AC and CQ are the required lines. 

Prop. XVIII. — From a given point A, draw a line that shall be 
equally distant from two given points B and C. 

Solution. — Let A be the first point, and let B and C be the other 
given points. 

Draw BC, bisect it in O, and then 
draw AO. From B and C, draw BQ 
and CP perpendicular to AO. 

The triangles BOQ and COP are 
equal in all their parts ; hence BQ equals CP. The line AO is there- 
fore the required line. 

Prop. XIX. — Through a given point draw a line cutting the sides 
of a given angle and making the interior angles equal to each other. 

Solution. — Let P be the given point and DAF the given angle. 

Draw the bisectrix AS, of the given 
angle, and through P draw PR perpen- 
dicular to AS, cutting the sides of the 
angle in Q and R. 

In the triangles ASQ and ASR, we 
have, AS common, the angles SAQ and 



A 



Q 



o 





16 



KEY. 




SAR equal by construction, and the angles ASQ and ASR equal, 
because they are right angles ; hence, these triangles are equal in all 
their parts. The angles SQA and SRA, are therefore equal, and con- 
sequently PR is the required line. 

Prop. XX. — Draw a line PQ parallel to the base BC of a trian- 
gle ABC, so that PQ shall be equal to the sum of BP and CQ. 

Solution. — Let ABC be the given triangle. 

Draw the bisectrices BR and CR, of the 
angles B and C, and through their intersection 
R, draw PQ parallel to BC. 

Since PQ is parallel to BC, the angles QRC 
and RCB are equal ; but, RCQ and RCB are 
equal by construction ; hence, the angles QRC and QCR are equal ; 
the triangle RQC is therefore isosceles (Bk. II. Prop. 12), and conse- 
quently RQ is equal to QC. In like manner it may be shown that 
RP is equal to PB. Hence, 

PR + RQ or PQ = BP + QC ; 

the line PQ is therefore the required line. 

Prop. XXI. — In a given isosceles triangle draw a line that shall 
cut off a trapezoid whose base is the base of the given triangle and 
whose other sides shall be equal to each other. 

Solution. — Let ADF be the given isosceles triangle. 

Draw the bisectrix FP of the angle F, 
and through the point in which it meets AD 
draw PQ parallel to DF. 

It may be shown as in the last problem, 
that the triangle PQF is isosceles, the side 




PROPOSITIONS FROM LEGENDRE. 17 

PQ being equal to QF. Since PQ is parallel to DF, the angles AQP 
and APQ are equal, and consequently AP is equal to AQ ; AD — AP 
or PD, is equal to AF — AQ or QF. Hence, PQ = QF = PD, 
and consequently PQ is the required line. 

Prop. XXII. — If two opposite sides of a parallelogram are bisected 
and lines be drawn from the points of bisection to the vertices of the 
opposite angles, show that these lines will divide the diagonal which 
they intersect, into three equal parts. 

Demonstration. — Let ABCD be any parallelogram, and let P and 
Q be the middle points of BC and AD. 

Draw PD and BQ, also draw QT 
parallel to AC. 

Since BP is parallel and equal to QD, 
BQ and PD are also parallel and equal 
(Bk. I. Prop. 30). In the triangles ARQ 

and CSP, the angles ARQ and CSP are equal because they are alter- 
nate exterior angles with respect to BQ and PD. The angles RAQ 
and SCP are equal, because they are alternate angles, and the sides 
AQ and CP are equal because they are halves of equal lines (Ax. 7) ; 
hence, the triangles are equal in all their parts : AR is therefore 
equal to SC. In the triangles ARQ and QTD, the angles ARQ and 
QTD are equal because their sides are parallel and lie in the same 
direction (Bk. I. Prop. 24) ; the angles RAQ and TQD are equal 
because they are opposite exterior and interior angles, with respect to 
BQ and PD, and the sides AQ and QD are equal by hypothesis; 
hence the triangles are equal in all their parts : QT is therefore equal 
to AR : but QT is equal to RS ; hence, QT = AR = SC, which was 
to be .proved. 




18 



KEY. 



Prop. XXIII. — Construct a triangle, having given the two angles 
at the base and the sum of the three sides. 



Solution. — Let AB be equal to the sum of the three sides, and 

G 




let the angles BAC and ABC be 
equal to the angles at the base of 
the required triangle. 

Draw the bisectrices AT and BT, 
of the angles A and B, and through 
their point of intersection T, draw 
TR parallel to CA or TS parallel to CB. 

The angles RTA and TAC are equal because they are alternate ; 
but TAC is equal to ATR by construction ; hence, the triangle ART 
is isosceles, and consequently AR = RT. In like manner BS = ST. 
The angle TRS is equal to CAS, the angle TSR is equal to CBA and 
the sum of RT, RS, and ST is equal to AB ; hence, TRS is the re- 
quired triangle. 



Prop. XXIV. — Construct a triangle, having given one angle, one 
of its including sides and the sum of the other two sides. 

Solution. — Let ABC be the given angle, AB one of its including 
sides, and let BD be equal to the sum A 

of the other two sides. 

Draw AD. At A make the angle 
DAC equal to the angle ADC (Bk. III. 
Prob. 4). 

The triangle ACD is isosceles and consequently, AC = CD (Bk. I. 
Prop. 12). Hence, BC + CA = BC + CD = BD. The triangle 
ABC is therefore the required triangle. 




PROPOSITIONS FROM LEGENDRE. 



19 




Prop. XXV. — Construct an equilateral triangle, having given one 
of its altitudes. 

Solution. — Let AD be equal to the given altitude. Draw any 
line AQ, perpendicular to AD, and on any 
part of it, as AB, construct an equilateral 
triangle ABC. Through D, draw DP par- 
allel to AQ, intersecting AC, produced in 
P. Through P draw PQ parallel to CB: 
then will APQ be the required triangle. 

For, the triangles ACB and APQ, being- 
similar, and ACB equilateral, APQ is also 

equilateral. But the altitude of the triangle APQ is equal to the 
given altitude : hence, APQ is the required triangle. 

Prop. XXVI. — Show that the three altitudes* of a triangle all 
intersect in a common point. 

Demonstration. — Let ABC be the given triangle. 

Through A, draw PQ parallel to CB, 
through B, draw RQ parallel to CA, and 
through C, draw PR parallel to AB. 
Through A, B and C, draw perpendicu- 
lars to PQ, QR, and RP, meeting the 
sides of the given triangle in G, F, and E. 

Now, BQ and CA are equal, because 
they are parallels between parallels ; for a like reason RB and AC are 
equal : hence, B is the middle point of QR. In like manner, C is the 
middle point of RP, and A is the middle point of PQ. Because PQ, 
QR, and RP are chords of the circle that circumscribes the triangle 
PQR, the perpendiculars, AG, BF, and CE, pass through the centre O 





20 KEY. 

of that circle : but these perpendiculars are also the altitudes of the 
given triangle ; hence, these altitudes all intersect at the same point, 
which was to be proved. 

Prop. XXVII. — If one of the acute angles of a right-angled tri- 
angle is double the other, shoiv that the hypothenuse is double the 
smaller side about the right angle. 

Demonstration. — Let ABC be a right-angled triangle in which 
the angle C is double the angle A. 

Draw the bisectrix CO of the angle 
C, and from the point O in which it 
meets BA, draw OP perpendicular to 
the hypothenuse CA. 

Because the angle C is equal to 
twice the angle A, we have the angle OCA equal to the angle OAC ; 
that is, the triangle OCA is isosceles ; hence, the perpendicular OP, 
bisects the side CA; consequently CP is equal to one-half of the 
hypothenuse. But CP is equal to CB, because the right-angled 
triangles CPO and CBO have the same hypothenuse CO, and the 
acute angle OCP in the one, equal to the acute angle OCB in the 
other. Hence, CA = 2CP = 2CB, which was to be proved. 

Prop. XXVIII. — Let a median be drawn from the vertex of any 
angle A of a triangle ABC; then show that the angle A is a right 
angle luhen the median is equal to half the side BC, an acute angle 
when the median is greater than half of BC, and an obtuse angle when 
it is less than half of BC. 

Demonstration. — Let ABC be-a triangle, 
and let AD be the median drawn from the 
vertex A. 




PEOPOSITIONS FEOM LEGENDEE. 21 

1°. Suppose AD = BD = DC. 

The triangle ADC is isosceles, because its sides AD and DC are 
equal ; hence, the angles DAC and DCA are equal. In like manner 
it may be shown that the angles DAB and DBA are equal. Hence, 
the sum of the angles CAD and DAB, or the angle BAC, is equal to 
the sum of the angles ABD and ACD, that is, the angle at A is equal 
to half the sum of the three angles of the given triangle, or to one 
right angle, which ivas to be proved. 

2°. Suppose AD > BD and consequently AD > CD. 

Since, AD > BD, we have from the converse of Prop. 9, Bk. I, 
the angle B greater than the angle BAD. In like manner the angle 
C can be shown to be greater than the angle DAC. Hence, the sum 
of the angles BAD and DAC, or the angle at A, is less than half the 
sum of the three angles of the given triangles, and consequently it is 
acute, which was to be proved. 

3°. Suppose AD < BD, and consequently AD < DC. 

It follows from the converse of Prop. 9, Bk. I, that the angle B 
is less than the angle BAD ; and the angle C is less than the angle 
CAD ; hence, the angle at A is greater than half the sum of the three 
angles of the given triangle, that is, it is obtuse, which was to be 
proved. 

Prop. XXIX. — Let any quadrilateral be inscribed about a circle: 
then shoiv that the sum of two opposite sides is equal to the sum of the 
other tv)o opposite sides. 

Demonstration. — Let ABCD be any quadrilateral circumscribed 
about the circle 0, and let its sides be tangent to this circle at the 
points P, Q, R, and S. 



22 



KEY. 



From the principle demonstrated in the 
Corollary to Prob. 14, Bk. Ill, we have the 
following equations : 



DS 

cs 

AQ 
BQ 



DP 
CR 
AP 
BR. 




Adding these equations, member to member, and recollecting that 
DS + CS = DC, AQ + BQ = AB, DP -J- AP = DA, and 
CR -f BR = CB, we have, 

DC + AB = DA + CB, 

which was to be proved. 

Prop. XXX. — Draw a straight line tangent to two given circles. 

Solution. — Let A and B be the centres of the two circles, and 
let AP and BQ be their radii, and 
let AP > BQ. 

Draw the line AB, through their 
centres. 

With a radius AR, equal to 
AP — BQ, and a centre A, de- 
scribe the circle AR ; through B draw BR tangent to this auxiliary 
circle, and let AR be drawn through the point of contact R, and 
prolonged to T ; draw the radius BL, parallel to AT, and through the 
points T and L draw the line TL. 

Because AR is perpendicular to BR (Bk. Ill, Prop. 9), and BL 
is parallel to AR, the angles TRB- and LBR are right angles. But, 
RT is equal to BL, by construction ; hence, the figure TRBL is a 
rectangle, and consequently the line TL is perpendicular to the radii 




PROPOSITIONS FROM LEGEKDRE. 23 

AT and BL, at their extremities ; TL is therefore tangent to both the 
o-iven circles ; hence, TL is the required line. 

Scholium. — A second tangent may, in like manner, be constructed 
on the other side of the circles, and it may be shown that both inter- 
sect the line of centres at a common point, to the right of B. 

By using an auxiliary circle whose radius is equal to AP -f BQ, 
two other tangents may be constructed intersecting each other on the 
line of centres, between A and B. The first pair of tangents is said 
to be external, the second pair is said to be internal. 

There may be several positions of the given circles. 

1°. If the circles are external to each other, they will always have 
two external and two internal tangents, common to both. 

2°. If the circles touch each other externally, they will have two 
external tangents and only one internal tangent. 

3°. If the circles cut each other, they will have two external tan- 
gents but no internal tangent. 

4°. If the circles touch each other internally, they will have one 
external but no internal tangent. 

5°. If one circle lies wholly within the other, they can have no 
common tangent whatever. 

Prop. XXXI. — Through a given point P, draw a circle that shall 
be tangent to a given line CB, at a given point B. 

Solution. — Let P be a given point, CB a given line, and B a 
given point on that line. 

Draw PB and bisect it by a perpendicular 
OC, meeting BC in C ; at B erect BO perpen- 
dicular to CB, and meeting OC in O. Then, 
with as a centre, and OB as a radius describe 
a circle. 




24 



KEY. 



Because OD is perpendicular to PB, at its middle point, is 
equally distant from B and P (Bk. I, Prop. 16) ; hence, the circle 
passes through P. Because BC is perpendicular to the radius OB, 
at its extremity, it is tangent to the circle at B. Hence, the circle 
BDP is the required circle. 

Prop. XXXII. — Let two circles intersect each other, and through 
either point of intersection let diameters of the circles be draivn : then 
show that the other extremities of these diameters and the other point 
of intersection, lie in the same straight line. 

Demonstration. — Let C and D be the centres of two circles 
which intersect each other at E and S, 
and let EQ and EP be the diameters 
of the circles through E. 

Draw QS, PS, and SE. 

Because ECQ is a diameter of the 
circle C, the angle QSE is a right angle 

(Bk. Ill, Prop. 18, Cor. 2); in like manner the angle ESP is a right 
angle : hence, the line ES meets QS and SP, making the sum of the 
angles ESP and ESQ equal to two right angles ; the lines QS and SP 
therefore form a single straight line QSP (Bk. I, Prop. 3), which was 
to be proved. 

Prop. XXXIII. — Through two given points, A and B, draw a 
circle that shall be tangent to a given line CP. 

Solution. — Let A and B be the given points and CP the given line. 

Draw AB, and prolong it till it 
intersects CP at C ; lay off BQ equal 
to AC, and on CQ, as a diameter, con- 
struct a semicircle QDC; draw BD 
perpendicular to CQ, cutting this semi- 





PROPOSITIONS FROM LEGENDRE. 



25 



circle in D; make CP = BD and draw PO perpendicular to CP; 
also draw GO perpendicular to, and bisecting AB ; from the point O, 
in which the last two lines intersect, as a centre, and with a radius 
equal to OP draw a circle BPA. 

The circle BPA is tangent to the line PC, at P, because CP is 
perpendicular to the radius OP, at its extremity. Since BD is perpen- 
dicular to CQ, it is a mean proportional between CB and BQ, that is; 
between CB and CA (Bk. IV, Prop. 23) ; hence, the points A and B 
are on the circle APB (converse of Prop. 30, Bk. IV) : consequently 
the circle BPA is the required circle. 



Prop. XXXIV. — Draw a circle that shall be tangent to a given 
circle C and also to a given line BP, at a given point P. 

Solution. — Let C be the given circle, BP the given line, and P 
the given point. 

Through P draw the line QO, perpen- 
dicular to BP, and make PQ equal to the 
radius CD of the circle C ; draw QC, and 
bisect it by a perpendicular SO ; from the 
point 0, in which SO and QO intersect, as 
a centre, and with a radius OP, describe 
the circle PD. 

Because SO is perpendicular to QC, at its middle point, OQ = OC 
and consequently OP = OD. But BP is tangent to the circle O, at 
P, because it is perpendicular to the radius OP at its extremity. The 
circle is tangent to the circle C, because the distance OC = OD -f- 
DC (Bk. Ill, Prop. 14, Scho.). Hence, the circle is the required, 
circle* 




26 



KEY. 



Prop. XXXV. — Draw a circle that shall be tangent to a given line y 
TP, and also to a given circle C, at a given point Q. 

Solution. — Let TP be the given line, C the centre of the given 
circle, and let Q be any given point on 
the circle C. 

Through Q draw the line CQO, and 
through the same point Q draw QT 
perpendicular to CO ; draw the bisec- 
trix of the angle QTP, meeting CO in 
O ; draw OP perpendicular to TP, and 
then with O as a centre and with OP as a radius, describe the circle O. 

Since O is on the bisectrix TO, OP and OQ are equal ; hence, 
the circle whose radius is OP, passes through Q, and is tangent to the 
circle C at Q ; it is also tangent to TP ; hence, it is the required 
circle. 




Prop. XXXVI. — Draw a circle that shall pass through a given 
point Q, and be tangent to a given circle C, at a given point P. 

Solution. — Let Q be the given point, C the centre of the given cir- 
cle, and P the given point on that circle. 

Draw PQ, and bisect it by the per- 
pendicular DO ; draw CP, and prolong 
it till it meets DO, in O ; with O as a 
centre and OP as a radius, describe the 
circle OP. 

Because DO is perpendicular to PQ, at its middle point, OQ ±= OP ; 
hence, the circle OP passes through Q. Because CO = CP -f- PO 
the circle OP is tangent to the circle C. Hence, the circle OP is the 
required circle. 




PROPOSITIONS FEOM LEGENDRE. 



27 



Prop. XXXVII. — Draw a circle, with a given radius, thai shall 
be tangent to a given line DP, and to a given circle C. 

Solution. — Let DB be the given radius, DP the given line, and 

C the centre of the given circle, 
and CT its radius. 

Draw a line BO, parallel to 
DP, and at a distance from it 
equal to DB. With C as a cen- 
tre, and a radius CO, equal to 
CT -j- DB, describe an arc cut- 
ting BO in the points B and O. With B as a centre, and with a 
radius equal to BD, describe a circle ; also with O as a centre, and 
the same radius, describe a second circle. 

These circles are tangent to the line DP, at the points D and P ; 
they are also tangent to the given circle, because the distance from C, 
to the centre of either, is equal to the sum of the radii CT and BD ; 
hence, either of the circles B and O, is the required circle. 




Prop. XXXVIII. — Find a point, in the prolongation of any diam- 
eter of a given circle, such that a tangent from it to the circumference 
shall be equal to the diameter of the circle. 

Solution. — Let C be the centre of the given circle, and let AB be 
any diameter. 

At B, draw BP perpendicular to AB 
and equal to the diameter AB ; draw 
PC, and at the point Q, in which it cuts 
the circumference, draw the tangent QT 
and prolong it till it meets the prolong- 
ation of AB, in T. 




28 KEY. 

The triangles CQT and CBP have the common angle PCT; the 
angles CQT and CBP are equal, because they are right angles ; and 
the sides CQ and CB are equal, because they are radii of the same 
circle : hence, these triangles are equal in all their parts, and conse- 
quently, QT = BP = AB ; T is therefore the required point. 

Prop. XXXIX. — Show that when two circles intellect each other, 
the longest common secant that can be drawn through either point of 
intersection is parallel to the line joining the centres of the circles. 

Demonstration. — Let C and D be the centres of two circles 
intersecting each other at S and R, 
and let KT be a common secant 
drawn through one of the points 
of intersection S. 

Draw CD ; and from C and D 
let fall CQ and DP perpendicular 
to KT, and meeting it in the points Q and P. 

The distances KQ and QS are equal (Bk. Ill, Prop. 6), as are also 
the distances SP and PT; hence, KT = 2QP : but QP is the projec- 
tion of CD on the line KT, and this projection is the greatest possible 
when KT is parallel to CD (Prop. VI, Key). Hence, the whole line 
KT is the greatest possible when it is parallel to CD, which was to be 
proved. 

Prop. XL. Problem. — Construct the greatest possible equilateral 
triangle whose sides shall pass through three given points, A, B, and 
C, not in the same straight line. 

Solution. — Let A, B, and C be the three given points. 
Join the three points, forming the triangle ABC ; on the sides AB 
and AC, as chords, construct segments of circles APB and AQC capa- 




PROPOSITIONS FROM LEGEKDRE. 



29 




ble of containing an angle equal to two-thirds of a right angle 

(Bk. Ill, Prob. 16) ; through A draw QAP 

parallel to the line through D and E, the 

centres of the segments ; from the points Q 

and P, draw.QC and PB, and prolong them 

till they meet at R. 

Any triangle whose sides pass through 
A, B, and C, and whose two vertices are in 

the arcs APB and AQC, is equilateral, because each of its angles is 
equal to two-thirds of a right angle ; but no other triangle whose sides 
pass through A, B, and C, can be equilateral. Now, of the group of 
equilateral triangles that can be formed, as just indicated, that one 
will be the greatest possible which has one of its sides the greatest 
possible ; but from Prop. XXXIX, Key, the side through A will be 
greatest possible when it is parallel to the line joining D and E: 
hence, QPR is the required triangle. 



Prop. XLI. — Show that the bisectrices of the four angles of any 
quadrilateral intersect in four points, all of which lie on the circum- 
ference of the same circle. 

Demonstration — Let ABCD be any quadrilateral, and let AR, 
BR, CP and DP, be the bisectrices 
of its four angles. Draw PR. 

The angle PSR, or its equal ASD, 
is equal to one-half the sum of ABC 
and DCB (Prop. Ill, Key). In like 
manner it may be shown that the 
angle PQR is equal to one-half the sum of the angles BAD and CD A, 
Hence, the sum of the angles PSR and PQR is equal to one-half the 




30 



KEY. 



sum of the four angles of the given quadrilateral, that is, to two right 
angles. Consequently, from the converse of Prop. 18, Bk. Ill, Cor. 4, 
the quadrilateral SPQR can be inscribed in a circle, which was to be 
proved. 



Prop. XLII. — If two circles touch each other externally, and if 
two common secants are drawn through the point of contact and ter- 
minating in the concave arcs, show that the lines joining the extremities 
of these secants, in the two circles, are parallel. 

Demonstration. — Let O and P be the centres of two circles 
which are tangent to each other at the 
point Q. 

Draw the secants BC and DE, ter- 
minating in the concave arcs BD and 
EC ; draw also BD and EC, and at Q 
draw TS tangent to both circles. 

The angle SQE is equal to QCE 
(Bk. Ill, Prop. 21 and 18) ; in like manner the angle TQD is equal to 
QBD. But SQE and TQD are equal because they are opposite 
angles ; hence, QCE and QBD are equal, and consequently, the lines 
CE and BD are parallel (Bk. I, Prop. 19, Cor. 1); which ivas to be 
proved. 




Prop. XLIII. — Let an equilateral triangle be inscribed in a circle, 
and let two of the subtended arcs be bisected by a chord : then show 
that the sides of the triangle divide the chord into three equal parts. 

Demonstration. — Let ABC be an equilateral triangle inscribed in 
a circle ASBCP, and let P and S be the middle points of the arcs 
APC and ASB. 



PROPOSITIONS FROM LEGENDRE. 



31 



Draw PS, cutting AC and AB in Q and R ; also draw AP and 
AS. 

The angles ASR and RAS are equal be- 
cause they are measured by halves of the 
equal arcs AP and SB ; hence, AR = RS. 
In like manner it may be shown that AQ — 
PQ. In the triangle AQR, the angle ARQ 
is measured by \ (SB -f PA), the angle 
AQR is measured by -J- (AS -f PC), and the angle QAR is measured 
by JCB, hence, the three angles are equal, and consequently QR = 
AQ = AR, or PQ = QR = RS, which was to be proved. 




Prop. XLIV. — Find a point within a triangle, such that the angles 
formed by drawing lines from it to the three vertices of the triangle, 
shall be equal. 

Solution. — Let ABD be the given triangle. On AB construct 
an arc that will contain an angle equal to 
four-thirds of a right angle ; on BD con- 
struct another arc that will contain an 
angle equal to four-thirds of a right angle, 
and intersecting the first at O ; draw OD, 
OA, and OB. 

The angles AOD and DOB are each 
equal to four-thirds of a right angle, and consequently, the angle AOB 
is also equal to four-thirds of a right angle ; hence, O is the required 
point. 

Prop. XLV. — Inscribe a circle in a quadrant of a given circle. 
Solution. — Let CBPD be the given quadrant. 




32 



KEY. 




Draw the bisectrix CP of the angle DCB, and at P draw a tangent 
PT to the quadrant, meeting CB pro- 
duced at T. 

Draw the bisectrix TO, of the 
angle CTP, and from the point O, in 
which it meets CP, draw OR and OQ 
perpendicular to CB and CD ; with 
O as a centre, and OR as a radius, 
describe a circle. 

Because TO is the bisectrix of the angle RTP, OR = OP 
(Prop. IV, Key) ; because CP is the bisectrix of the angle BCD, 
OR = OQ : hence, the circle OR passes through P and Q. This 
circle is tangent to CT because CT is perpendicular to OR at R, and 
for a like reason it is also tangent to CD at Q ; the circle OR is tan- 
gent to the arc DPB, because CO = CP — OP : hence, the circle 
OR is the required circle. 

Prop. XLVL — Through a given point P, within a given angle 
ABC, draio a circle that shall be tangent to both sides of that angle. 

Solution. — Let P be the given point and ABC the given angle. 

Draw the bisectrix BO, of the 
given angle, and also the line BP; 
from any point Q, of BO, draw QR 
perpendicular to AB, and from Q, 
as a centre, and with a radius QR, 
draw the arc RS, cutting BP in S ; 
draw SQ and also draw PO parallel 

to SQ, intersecting BO at O ; then draw OA parallel to QR, and with 
O as a centre and OA as a radius, describe the circle APC : also draw 
OC perpendicular to BC. 




PROPOSITIONS FROM LEGEKDRE. 33 

The triangles BSQ and BPO have the angle at B common, and 
since QS is parallel to OP, the angles BQS and BOP are equal, as are 
also the angles BSQ and BPO; hence, the triangles are similar 
(Bk. IV, Prop, 18), and consequently, their corresponding sides are 
proportional. 

We have, therefore, the proportion 

BQ : BO : : QS : OP . . . (1). 

In like manner, we have from the triangles BQR and BOA, the 

proportion, 

BQ : BO : : QR : OA . . . (2). 

Because the first three terms of proportions (1) and (2) are equal, 
each to each, their fourth terms must be equal, that is, OP = OA. 
Hence, the circle whose radius is OA, passes through P. The distance 
OC is equal to OA (Prop. IV, Key) ; hence, the circle whose centre 
is O passes through C. Furthermore, BA is perpendicular to OA, at 
its extremity, and BC is perpendicular to OC, at its extremity; 
hence, the circle whose radius is OA, is tangent to both sides of the 
given angle ; it is therefore the required circle. 

Prop. XL VII. — Show that the middle points of the sides of any 
quadrilateral are the vertices of an inscribed parallelogram. 

Demonstration. — Let ABCD be any quadrilateral, and let P, Q, 
R, and S, be the middle points of its four 
sides. 

Draw PQ, QR, RS, and SP ; also draw 
the diagonals BD and AC. 

Because AQ = QB, and AP = PD, 
we have the proportion, 

AP : PD : : AQ : QB. 




34 



KEY. 



Hence, from Bk. IV, Prop. 16, PQ is parallel to DB. In like manner 
it may be shown that RS is parallel to DB : hence, PQ and SR are 
parallel to each other. The lines QR and PS are parallel to AC, for 
similar reasons ; hence, they are parallel to each other, and conse- 
quently, the figure PQRS is a parallelogram inscribed in the given 
quadrilateral, which was to be proved. 




Q M 



Prop. XLVIII. — Inscribe in a given triangle, a triangle whose sides 
shall be parallel to the sides of a second given triangle. 

Solution. — Let ABC be the first given triangle, and let DEF be 
the second given triangle. 

Take any point P, on the side AB, and 
through it draw PQ parallel to DE ; from 
Q draw QR parallel to EF, and from P 
draw PR parallel to DF, intersecting PR in 
R ; draw BR and prolong it to S. Through 
S draw SM parallel to RQ, and SN parallel 
to RP, and then join M and N. 

From the triangles BSN and BRP,we have (Bk. IV, Prop. 15), 

BR : RS : : BP : PN . . . (1). 
From the triangles BSM and BRQ, we have, like reasons, 

BR : RS : : BQ : QM . . . (2). 
Combining proportions (1) and (2), we have, 
BP : PN : : BQ : QM; 
hence, from (Bk. IV. Prop. 16), we have, NM parallel to PQ. The 
sides of the inscribed triangle NSM,-are therefore parallel to the sides 
of the triangle PRQ, and consequently, to the sides of the triangle 
DFE. The triangle NSM is therefore the required triangle. 



PROPOSITIONS FROM LEGEKDKE. 



35 




Prop. XLIX. — Through a point P, within a given angle, draw a 
line such that it and the parts of the sides that are intercepted shall 
contain a given area. 

Solution. — Let BAG be the given angle, and P the given point. 

Through P, draw DPF parallel to AC, 
and complete the parallelogram DFGA, in 
such manner that its area shall be equal to 
the given area. To do this, first construct 
a triangle equal to the given area (Bk. IV, 
Prob. 6), then convert this into an equiva- 
alent triangle whose altitude is equal to the 

altitude DE, and then make DF and AG each equal to one-half the 
base of the last triangle. From G, draw GH perpendicular to AC 
and make it equal to PD ; with H as a centre, and a radius equal to 
PF describe an arc cutting AC in C ; then draw CP and prolong it 
toB. 

The triangles DBP, PFI, and IGC are similar (Bk. IV, Prop. 21); 
they are therefore proportional to the squares of their homologous 
sides DP, PF, and GC ; hence, we have the continued proportion 

PFI : DBP : GIC : : PF 2 : PD 2 : GC 2 . . . (l), 
whence, by composition, (Bk. II, Prop. 6), 



PFI : DBP + GIC : : PF 



PD 2 + GC 2 



(2). 



But, from the nature of the construction (Bk. IV, Prop. 11), we have, 
HC 2 or PF 2 = HG 2 -f GC 2 = PD 2 + GC 2 ; 

hence, the terms of the second couplet of proportion (2) are equal, 
and consequently the terms of the first couplet are equal, that is, 

PFI = DBP -4- GIC. 



36 KEY. 

If now we take away from the parallelogram AF, the triangle PFI 
and then add to it its equal, DBP + GIC, we shall have the triangle 
BAC. The triangle BAC is therefore equal to the parallelogram AF, 
which is equal to the given area ; hence, BAC is the required triangle. 

Prop. L. — Construct a parallelogram, whose area and perimeter 
are respectively equal to the area and perimeter of a given triangle. 

Solution. — Let ABC he the given triangle. Prolong AB, mak- 
ing BD = BC, and bisect AD in E; 
draw BF parallel to AC; with A as a 
centre and a radius equal to AE, describe 
an arc cutting BF in G ; draw AG ; bi- 
sect AC, in II, and complete the parallel- 
ogram AHFG. 

The sides AG and HF of the parallel- £~ Jf~ ~^r 

ogram AF, are each equal to AE, and 

consequently their sum is equal to AD, or AB -f BC. The sides AH 
and GF are each equal to one-half of AC, and consequently their sum 
is equal to AC ; hence, the perimeter of the parallelogram AF is equal 
to that of the given triangle. The altitude of the parallelogram is 
equal to that of the given triangle, and the base of the parallelogram 
is one-half that of the given triangle ; hence, the area of the parallel- 
ogram is equal to that of the given triangle (Bk. IY, Prop. 5 and 6). 
The parallelogram AGFH is therefore the required parallelogram. 

Prop. LI. — Inscribe a square in a semicircle, that is, a square, 
two of whose vertices are in the diameter and the other two in the semi- 
circumference. 

Solution. — Let ADCB be a semicircumference whose diameter is 
AB, and whose centre is O. 




PROPOSITIONS FROM LEGENDRE. 



37 




Draw AE perpendicular to, and equal to, AB ; draw EO, cutting 
the semicircumference in D ; draw DC -^ 

parallel to AB ; also, draw DF and CG per- 
pendicular to AB ; draw OC. 

The triangles OAE and OFD are similar ; 
but, OA = -|-AE, and consequently OF == 
•iFD. The triangles OFD and OGC are 
equal in all their parts ; hence, OG- =± OF, 
and consequently FG = FD ; but FD = GC, and FG = DC ; hence, 
FDCG is the required square. 

Prop. LII. — Through a given point P, draw a line cutting a tri- 
angle, so that the sum of the perpendiculars to it from the two vertices, 
on one side of the line, shall be equal to the perpendicular to it, from 
the vertex, on the other side of the line. 

Solution. — Let P be the given point and ABC the given triangle. 

Draw the median AD, and on it 
lay off DO == £DA; draw PO, and 
prolong it till it meets DC prolonged, 
in S ; upon this line let fall the per- 
pendiculars CG, AF, DK and BE. 

Because D is the middle of CB, 
we have the following equations : 

SD = SC + |CB 
SD = SB — |CB. 

Adding, and dividing by 2, we have, 

SD = i(SC + SB) . . . (1). 

From the similar triangles SCG and SBE, we have, 

SC : SB :: CG : BE . . . (2); 




38 KEY. 

whence, by composition 

SC : SC + SB : : CG : CG + BE . . . (3). 

From the similar triangles SCG and SDK, we have 

SC : SD : : CG : DK . . . (4). 

Because the antecedents in (3) and (4) are equal, each to each, 
we have, 

SC + SB : SD : : CG + BE : DK . . . (5). 

But, SC + SB = 2SD, hence, 

CG + BE = 2DK . . . (6). 
From the similar triangles AFO and DKO, we have, 
AF : DK : : AO : DO . . . (V) ; 
but, AO = 2DO by construction, hence, 

AF = 2DK . . . (8). 
From equations (6) and (8), we have, 

AF = CG + BE; 
hence, PS is the required line. 

Prop. LIII. — Shoio that the line which joins the middle points of 
two opposite sides of any quadrilateral bisects the line joining the 
middle points of the two diagonals. 

Demonstration. — Let ABCD be any quadrilateral, F and H the 
middle points of two opposite sides ; and 
let L and K be the middle points of the 
two diagonals BD and AC. 

Draw LK, FH, FL and KH. " 

Because F is the middle point of AD, 
and L the middle point of BD, FL is 




PEOPOSITIONS FEOM LEGEKDEE. 39 

t 

parallel to AB, and equal to f AB. In like manner it may be shown, 
that HK is parallel to AB, and equal to JAB. Hence, HK and LF 
are parallel and equal to each other. The triangles FOL and HOK, 
have their angles equal each to each and the sides LF and KH equal; 
hence, they are equal in all their parts, and consequently, OL = OK, 
which was to be proved. 

Prop. LIV. — If from the extremities of one of the oblique sides of 
a trapezoid, lines be drawn to the middle of the opposite side, show that 
the triangle thus formed is equal to one-half the given trapezoid. 

Demonstration. — Let BPQR be a trapezoid, and let BD and PD 
be lines drawn from B and P, to D, the 
middle point of QR. 

Through D, draw TD parallel to BP, 
and prolong it till it meets the prolonga- 
tion of BR, in S. 

The triangles DRS and DTQ, have 
their corresponding angles equal and the sides DR and DQ equal ; 
hence, they are equal in all their parts. If we take the triangle DTQ 
from the given trapezoid, and then add the equal triangle DRS, we- 
shall have the parallelogram BPTS ; hence, this parallelogram is equal- 
to the given trapezoid. The parallelogram PTSB, and the triangle: 
PBD, have a common base BP and the same altitude ; viz., the per- 
pendicular distance from D, to PB ; hence, from Prop. 5 and 6, Bk. IV, 
the triangle is equal in area to one-half the parallelogram, and conse- 
quently, to one-half the given trapezoid, which was to be proved. 

Prop. LY. — Find a point in the base of a triangle, such that the 
lines drawn from it parallel to, and limited by, the other sides of the 
triangle, shall be equal to each other. 




40 



KEY. 



Solution. — Let ABC be the given triangle. Lay off AE equal to 
AC, and complete the parallelogram ACFE. 
Draw FA cutting BC in ; draw OP par- 
allel to FC, and OQ parallel to FE. 

From the similar triangles AFC and 
AOP, we have, 

AF : AO : : FC : OP . . . (1). 

From the similar triangles AFE and 
AOQ, we have, 

AF : AO : : FE : OQ . . 

From (1) and (2), 

FC : OP : : FE : OQ. 

But, FC = FE ; hence OP = OQ ; the point O is therefore the 
required point. 




(2)- 



Prop. LYI. — Show that the line drawn from the middle point of 
the base, of any triangle, through the middle point of any line of the 
triangle parallel to the base, will pass through the opposite vertex, if 
sufficiently produced. 

Demonstration. — Let ABC be any triangle, SR a line parallel to 
the base BC, and let F and P, be the middle 
points of BC and SR. 

Draw PF, and prolong it. 

The straight line drawn from A to F, 
bisects SR (Bk. IV. Prop. 22), that is, it 
passes through P ; but only one ^straight 

line can be drawn through F and P, consequently, the line FP, if 
sufficiently produced, will pass through A, which was to be proved. 





PROPOSITIONS FROM LEGENDRE. 41 

Prop. LVII. — Show that the three medians of any triangle meet 
in a common point. 

Demonstration. — Let ABC be any triangle and let BE and CD 
be two of its medians, intersecting each 
other in P. 

Draw DE ; and through F, the middle 
point of BC, draw FP, cutting DE in Q, 
and prolong it. 

The line DE is parallel to BC ; conse- 
quently the triangles PFB and PQE are similar, as are also the trian- 
gles PFC and PQD. From PFC and PQD, we have, 

PF : PQ : : FC : QD . . . (1). 
From the triangles PFB and PQE, we have, 

PF : PQ : : FB : QE . . . (2). 

From (1) and (2), we have, 

FC : QD : : FB : QE. 

But, FC = FB, hence, QD = QE; the line FP must therefore 
pass through the middle of DE, and therefore, from Prop. LVI, Key, 
it must also pass through A, that is, the line through A and P is the 
third median of the given triangle. Hence, the three medians pass 
through P, which was to be proved. 

Prop. LVIII. — On the sides AB and AC of any triangle ABC, 
construct any two parallelograms ABDE and ACFG ; prolong the 
sides DE and FG till they meet in H; draw HA, and on the third side 
of the triangle BC, construct a parallelogram two of whose sides are 
parallel and equal to HA : then show that the parallelogram on BC is 
equal to the sum of the parallelograms on AB and AC. 




42 KEY. 

Demonstration.— Let ABC be any triangle ; Let AD and AF be 
any parallelograms constructed on AB and 
AC as sides ; and let H be the point in 
which DE and FG meet, when prolonged ; 
let CP be a parallelogram whose sides BP 
and CQ are parallel and equal to HA. 

Draw HA, and prolong it to S ; also 
prolong PB and QC, to K and L. 

The parallelograms ABDE and ABKH 
have a common base AB, and a common 
altitude; hence, they are equal (Bk. IV, 

Prop. 1) ; the parallelograms ABKH and SPBT have equal bases, 
HA and ST, and a common altitude ; they are therefore equal : hence, 
the parallelogram ABDE is equal to the parallelogram STBP. In 
like manner it may be shown that the parallelogram ACFG is equal 
to the parallelogram STCQ. The sum of the parallelograms TP and 
TQ is equal to the parallelogram BCQP; hence, the parallelogram 
BCQP is equal to the sum of the parallelograms ABDE and ACFG, 
which was to be proved. 

Prop. LIX. — Assuming the principle demonstrated in the last 
proposition^ deduce from it the truth that the square on the hypothenuse 
of a right-angled triangle is equal to the sum of the squares on the 
other two sides. 

Demonstration.- — Let ABC be a triangle right-angled at A, and 
let BQ, AD, and AF be squares described on its sides. Let DE and 
FG, when prolonged, meet in H. 

Draw HA, and prolong it to S ; prolong PB to K, and QC to L. 

The right-angled triangle AGH has the side AG. = AC, and the 



PEOPOSITIONS FEOM LEGEKDEE. 



43 




side GH = AE == AB ; hence, it is equal to the triangle CAB in 
all its parts, that is, AH is equal to CB, * 

and the angle HAG is equal to the angle 
ACB. Since the angle CAG is a right 
angle, the sum of the angles HAG and 
CAT, or the sum of the angles ACB and 
CAT is equal to a right angle, and conse- 
quently the angle ATC is a right angle ; 
hence, TS is parallel to CQ and BP. The 
square CP, is therefore a parallelogram 
whose sides are parallel and equal to HA, 
and consequently, from Prop. LVIII, Key, it is equal to the sum of 
the squares AD and AF, which was to be proved. 

Prop. LX. — If from the middle of the base of a right-angled 
triangle a line is drawn perpendicular to the hypothenuse dividing 
it into two segments, show that the difference of the squares of these 
segments is equal to the square of the other side about the right angle. 

Demonstration. — Let ACB be a right-angled triangle, and D the 
middle of its base. 

Draw DE perpendicular to AB ; draw also 
the line AD. 

In the right-angled triangle ADE, we have, 

AD 2 = AE 2 + ED 2 . . . (1). 
In the right-angled triangle DEB, we have, 




DB 2 , or CD 2 = EB 2 + ED 2 

Subtracting (2) from (1), we have, 

AD 2 — CD 2 = AE 2 — EB 2 



00- 
(8). 



44 



KEY. 



But, AD — CD = AC , hence, from (3), we have, 



AE 



EB 2 = AC 2 , 



which was to be proved. 



Prop. LXI. — If lines are drawn from any point P to the four 
vertices of a rectangle, show that the sum of the squares of the two 
lines drawn to the extremities of one diagonal is equal to the sum of 
the squares of the two lines drawn to the extremities of the other 
diagonal. 

Demonstration. — Let ABDC be any rectangle, and P any point 
in its plane. 

Draw PA, PB, PC, and PD ; also draw 
the diagonals AD and CB, and through P 
draw EPF perpendicular to CD. 

From the right-angled triangles PEB 
and PFC, we have, 

PB 2 = PE 2 + EB 2 



E B 




F B 



PC 2 = PF 2 + FC 2 , 



whence, by addition, 



PB + PC = PE + PF + EB + FC . . . (1). 

From the right-angled triangles PEA and PFD, we have, 

PA 2 = PE 2 + EA 2 

PD 2 = PF 2 + FD 2 , 
whence, by addition, 



PA 2 + PD 2 = PE 2 + PF 2 + FD 2 + EA 2 . . . (2). 



PROPOSITIONS FROM LEGENDRE. 



45 



Because, EB = FD, and EA == FC, the second members of (1) 
and (2) are equal; and consequently the first members are also equal; 
hence, 

PB 2 + PC 2 = PA 2 + PD 2 , 

which was to be proved. 

Prop. LXII. — Let a line be drawn from the centre of a circle to 
any point of any chord ; then show that the square of this line, plus 
the rectangle of the segments of the chord, is equal to the square of the 
radius. 

Demonstration. — Let be the centre of any circle EBA, BA 
any chord, and P any point of that chord. 

Draw OP, and through P draw the chord 
ED perpendicular to OP ; also, draw EO. 

From the right-angled triangle OPE, we 
have, 



OE* 



OP 2 + EP 2 . . . (1). 




From Prop. 28, Bk. IV, we have, 

EP x PD, or EP 2 : 

substituting in (l), we have, 



BP x PA; 



OE 2 = OP 2 + BP x PA, 

which was to be proved. 

Prop. LXIII. — Draw a line from the vertex of any scalene triangle 
to a point in the base, such that this line will be a mean proportional 
between the segments, into which it divides the base. 

Solution. — Let ABC be a scalene triangle, AC its base, and B its 
vertex. Through A, B, and C pass a circle ABCG- and let its centre 



46 



KEY. 



be F ; draw the diameter BFG, and on the radius FB, as a diameter, 
describe a second circle cutting AC, in D 
and R ; draw BE, DF and EG. 

The angles BDF and BEG, being in- 
scribed in semicircles, are right angles, 
and consequently, the triangles BDF and 
BEG are similar; hence, BD ±= DE. 

From Prop. 28, Bk. IV, We have, 

BD x DE = AD x DC, or BD 2 = AD x DC; 
hence, BD is the required line. 

A second line could be found, by drawing a line from B to R, that 
would be a mean proportional between AR and RC. 




Prop. LXIV. — Show that the sum of the squares of the diagonals 
of any quadrilateral is equal to the sum of the squares of the four 
sides of the quadrilateral, increased by four times the square of the 
distance between the middle points of the diagonals. 

Demonstration. — Let ABCD be any quadrilateral, AC and BD 
its diagonals, and E and F the middle points 
of the diagonals. 

Draw EF, ED, EB ; FC and FA. 

From Prop. 14, Bk. IV, we have the fol- 
lowing relations : 




CD 2 + CB 2 



2BF 2 + 2CF 2 



AD 2 + AB 2 = 2BF 2 + 2AF 2 ; 



whence, by addition, 

CD 2 + CB 2 + AD 2 + AB 2 



4BF 2 + 2 (CF 2 + AF 2 ). 



PKOPOSITIONS FROM LEGENDRE. 



47 



But, from the same proposition, we have, 

CF 2 + AF 2 = 2 AE 2 -f 2EF 2 ; 
which gives, when substituted in the preceding equation, 

CD 2 + CB 2 + AD 2 + AB 2 = 4BF 2 + 4AE 2 + 4EF 2 . 

But, 4BF 2 = BD 2 , (Bk. IV, Prop. 8, Cor.); also 4AE 2 = AC 2 5 
hence, 

CD 2 + CB 2 _j_ AD 2 + AB 2 = BD 2 + AC 2 + 4EF 2 , 
which was to be proved. 



Prop. LXV. — Construct an equilateral triangle equal in area to 

any isosceles triangle. 

Solution. — Let ABC be any isosceles triangle. On AB, as a 
side, construct the equilateral triangle 
ABD ; draw the line DCE, which is the 
common bisectrix of the angles D and C. 

On DE, as a diameter, draw a semi- 
circle DFE, and from C erect the line CF 
perpendicular to DE ; draw EF ; and 
from E, as a centre, with EF as a radius, 
draw the arc FG-, cutting DE in G; 
through G, draw GP and GQ, parallel to DB and DA. Also draw FD. 

From Prop. 23, Bk. IV, Cor. 2, we have, 

EF 2 , or EG 2 = EC x ED, or 
ED : EG : : EG : EC . . . (1). 
From the similar triangles EDB and EGP, we have, 
ED : EG : : EB : EP . . . (2). 




48 KEY. 

From (1) and (2), we have, 

EG : EC : : EB : EP, or EG x EP = EC x EB ; 

but, EG x EP is equal to the area of the equilateral triangle QGP, 
and EC x EB is equal to the area of the isosceles triangle ACB ; 
hence, QGP is the required triangle. 

Prop. LXVI. — In a triangle ABC let two lines be drawn from the 
extremities of the base BC, intersecting at any point P on the median 
through A, and meeting the opposite sides in the points E and D; 
show that DE is parallel to BC. 

Demonstration. — Let ABC be the given triangle, P any point on 
the median, and CD, BE, lines drawn from 
C and B, through P. 

Through P, draw SPR parallel to BC; 
also, join D and E. 

From the similar triangles DSP and 
DBC, we have, 

DP : DC : : SP : CB . . . (1). 

From the similar triangles EPR and EBC, we have, 

EP : EB : : PR, or PS : BC . . . (2). 

From (1) and (2), we have, 

DP : DC : : EP : EB . . . (3). 

Whence, by division (Bk. II, Prop. 6), we have, 

DP : DC — DP : : EP : EB — EP, or 
DP : PB : : EP : PC. 

Hence, the triangles DPE and BPC are similar (Bk. IV, Prop. 20), 
and consequently, the angles DEP and PBC are equal ; hence, DE is 
parallel to BC, which ivas to be proved. 




PROPOSITIONS FROM LEGENDRE. 



49 



APPLICATIONS OF ALGEBRA TO GEOMETRY. 



Prop. LXVII. — In a right-angled triangle ABC, given the base 
BA, and the sum of the hypothenuse and perpendicular to find the 
hypothenuse and the perpendicular. 

Solution. — Denote BA by c, BC by x, AC by y, and the sum of 
BC and AC by s. 

Then, x -\- y = s . . . (1). 

From Bk. IV, Prop. 11, a* = y 2 -f c 2 . . . (2). 

From (1), we have, x = s — y. 




Squaring, 



— 5 2 _ 2sy + y 2 . . . (3). 



Subtracting (2) from (3), = s 2 — 2sy — c\ 

s 2 — c 2 
Transposing and dividing, y = — ; 

AS ' 
§2 — c 2 §2 _[_ c 2 

whence, x = s — = -^-. 

If c = 3 and s = 9, we have # = 5 and y = 4. 

Prop. LXYIII. — /n a right-angled triangle, given the hypothenuse 
and the sum of the sides about the right angle, to find these two sides. 

Solution. — Denote BC by a, BA by x, CA by y, and BA + AC 
by s. 

Then, x -f- y = s . . j (1) 

and # 2 -\- y 2 = a 2 . . . (2). 

From (1), 38 = s 2 — 2sy + y 2 . . . (3). 

3 



50 



KEY. 



Combining (2) and (3), 

a 2 — s 2 



whence, 
Solving (5), 



y 2 — sy — 



y — \8 ± V'W — i« 2 
Substituting in (1), 



x = U ^ Via 2 — i« 2 



(4), 
(5). 

• (6). 

: (7). 




If a = 5 and s = 7, we have, y = 4 and 3, # = 3 and 4. 

Prop. LXIX. — In a rectangle, given the diagonal and the peri- 
meter, to find the sides. 

Solution. — Denote AC by d, the side BA by x, the side BC by y, 
and the sum of BA and BC, or half the perimeter 
by a. 

Then, 

* -b V — a, 
and, 

# 2 + y* — cP, 
from which we obtain by solution, 

y = ia ± Vp - fr* 
x — ±a =F Vid* — \a\ 
If d = 10 and a = 14, we have y = 8 and 6, and # = 6 and 8. 

Prop. LXX. — Given the base and perpendicular of a triangle, to 
find the side of an inscribed square. 

Solution. — Let ABC be the given triangle, and let FGHE be the 
inscribed square. 



PEOPOSITIONS FEOM LEGENDEE. 



51 




Denote AB by b, CD by a, GH by x, 
whence CI is equal to a — x. 

From the similar triangles ACB and GCF, 

we have, AB : CD : : GF : CI, 

or, b : a : : x : a — x, 

ab 

whence, x = -. 

a -\- b 

The side of the inscribed square depends only on the base and 

altitude of the given triangle. 

Prop. LXXI. — In an equilateral triangle, given the distances from 
a point within to the three sides, to find one side of the triangle. 

Solution. — Let ACB be an equilateral triangle, D any point 
within it, DE, DG, and DF perpendiculars 
from D to the sides of the triangle and CH 
the altitude of the triangle. 
Draw DA, DB, and BC. 

Denote DG by a, DE by b, and DF by c ; 
also denote one of the sides of the triangle by 

2x, hence, AH = a; and CH = ^AC 2 — AH 2 = V^a* Z x \ or 

CH = x\/3. 

The area of the given triangle is equal to x x x^fz, or x 2 \/d ; 
the area of the triangle ADB is equal to a x # ; 
the area of the triangle BDC is equal to b x x ; 
the area of the triangle ADC is equal to c x #. 
But the last three triangles make up the first ; hence, 
x 2/ \/d = ax -f bx -j- cx r 
a -j- b -f- c 




whence, by solution, 



x = 



a/3 



52 



KEY. 



Prop. LXXII. — In a right-angled triangle, having given the base 
and difference between the hypothenuse and perpendicular, to find the 
sides. 



Let a = the base AB, 

b = the difference between AC and BC, 
x = the hypothenuse AC, and 
y = the perpendicular CB. 
Then, x — y = b, by the conditions, • • • 
x 2 = a 2 + y 2 (Bk. IV. Prop. 11), . . 

From equation (1) we have, 

x = y + b; 
by squaring, x 2 = y 2 -\~ 2by -f- b 2 . 

Substituting this value of x 2 in equation (2), 

y 2 + 2by -\- b 2 = a 2 -+- y 2 ; 

«2 _ 
hence, 26y = a 54 — 6 2 , or, y = 




b 2 



2b ' 

substituting this value in equation (1), we readily find 

a 2 -f b 2 



x = 



26 



Prop. LXXIII. — In a right-angled triangle, having given the 
hypothenuse and the difference between the base and perpendicular, to 
determine the triangle. 

Let a = the hypothenuse AC, 

b = difference between AB and BC, 



PROBLEMS FEOM LEGENDRE. 

x =: the base AB, and 
y = the perpendicular BC. 
Then, x — y = b, by the conditions, • • (1) 
a 2 = x 2 + y 2 , (Bk. IV, Prop. 11) . (2) 
From equation (1), we have, 

x — y + l\ 
x 2 = y 2 + 2by + b 2 . 
Substituting this value of x 2 in equation (2), we have, 

a 2 = y 2 + 2by + b 2 + y 2 -, 
and by transposing and reducing, 

V 2 + oy = — - — ; 




and by squaring, 



hence, 



y = 



b-\-y/%a 2 -b 2 ^ 



y 



l b j- 1 /2a 8 ~ & 2 \ 



- 



• 



Substituting either of these values in equation (1), we readily find 
the corresponding value of x. 

Note. — The positive value of the unknown quantity generally fulfils the con- 
ditions of the problem, understood in its arithmetical sense. 

The negative value will always satisfy the conditions of the equation: with 
its sign changed, it may be regarded as the answer to a problem which differs 
from the one proposed only in this : that certain quantities which we*e additive 
kave become subtractive, and the reverse. 

Prop. LXXIV. — Having given the area of a rectangle inscribed in 
a given triangle, to determine the sides of the rectangle. 

By a given triangle, we mean one whose sides are all known, or 
given. 

1st. To find the perpendicular and segments of the base: 



54 



KEY. 




Let ABC be a triangle, in which the three 
sides are given : viz., 

b = AB, a- AC, and c = BC. 
Let CD be drawn perpendicular to AB, and £ 

let y = CD, and x = AD ; then will b — x = DB. 

Then, a 2 = x 2 4- y 2 (1) 

c 2 = y 2 + x 2 — 2bx + b 2 • • • (2) ; 

substituting in equation (2) the value of x 2 + y 2 = a 2 , from 
equation (1), 

c 2 = a 2 — 2bx + b 2 ; hence, 
a? + &2 _ C 2 



rr = 



26 



and substituting this value in equation (1), we find the altitude of 
the triangle. 

2d. To find the sides of the rectangle. 
Suppose the rectangle to be inscribed in the 
triangle ACB. 

Let d — the area of the rectangle, 

x = its base, / 

y = its altitude, and 

h = CD, the altitude of the triangle. 




Then, by similar tr 
AB 
b 

and 



angles, 

CD : : GF : CJ; that is, 
h : : x : h — y ; hence, 
hx = bh-by • • • (1), 
xy = d. by the conditions, (2). 
d 



From equation (2), we have x = - ; 

* if 



PEOBLEMS FEOM LEGENDRE. 55 



substituting this value in equation (1), we have, 

h- = bh — by ; clearing of fractions, 
hd = bhy — by 2 ; or, 
y 2 — hy = — — ; whence, 



h 1 lb h 2 - 4hd h 1 lb h 2 - 4hd 

y = + 2 + 2V — i — *' y - + 2 - 2V — r — 

Substituting these values of y in equation (1), we find the corres* 
ponding values of x. 

Prop. LXXV. — In a triangle, having given the ratio of the two 
sides, together with both the segments of the base made by a perpendic- 
ular from the vertical angle, to determine the triangle. 

Let ACB be a triangle, and CD a line drawn perpendicular to 
the base AB. 

Let AC = x 

CB= y 
CD = z 
AD= a 
DB= b 
c = ratio. 

x 
Then, - = c. by the conditions • • • (1) 

y 

z 2 -a 2 = z 2 (Bk. IV., Prop. 11, Cor. 1) . • (2) 
y 2 - b 2 = z 2 (Bk. IV., Prop. 11, Cor. 1) • • (3). 
Subtracting (3) from (2), member from member, 

#a _ y i 4. b 2 - a 2 = ; or, x 2 = y 2 + a 2 — b 2 • (4). 




56 KEY. 

From equation (1) we have 

x = cy ; hence, x 2 = c 2 y 2 • • . (5) ; 
combining (4) and (5), we have 

C 2 y 2 _ y 2 _|_ a 2 _ £2 ? and ( C 2 _ J) y 2 _ a 2 __ g2 . 
a 2 _ £2 /«2 _ 52 

hence, y 2 = _____ . and y = ± y__. 

substituting these values in equation (1), we find the corresponding 
values of x. 

Prop. LXXVI. — In a triangle, having given the base, the sum of 
the other two sides and the length of a line drawn from the vertical 
angle to the middle of the base, to find the sides of the triangle. 

Let ABC be a triangle, and CD the line drawn from the vertex 
C to D, the middle point of the base. 

Let AC = x 

BC = y 

AD = 6 

CD =a 

AC -f- BC = s. 

Then, x -f y = s, by the conditions • • (1) 

and x 2 + y 2 = 26 2 + 2a 2 , (Bk. IV., Prop. 14) • • (2) ; 

equation (1), by transposing and squaring, gives 

y 2 = x 2 — 2sx 4 s s 2 ; 

substituting this value in equation (2), we have 

2x 2 - 2sx +■ s 2 =_ 26 2 + 2a 2 ; 




PEOBLEMS FROM LEGENDRE. 57 

transposing and reducing, we have 

26 2 + 2a 2 - s 2 
x l — sx = -. ; whence, 

s + a/46 2 + 4a 2 — s 2 , s— -t/4& 2 + 4a 2 — s 2 ' 
x = ^— - , and * = 2 Z 

Prop. LXXVII. — In a triangle, having given the two sides about 
the vertical angle, together with a line bisecting that angle and termi- 
nating in the base, to find the base. 

Let ACB be a triangle, and CD a line bisecting the angle ACB, 
and terminating in the base at D. 



Let 




AC =a 
BC = b 
CD = c 
AD = # 
DB =y. 






G 




AD B 


Then, x 


• y 


: : a : b 


(Bk. 


IV., 


Prop. 17) ; 


whence, 






bx 


= ay 


• • • (i); 



also ? a x b = c 2 + xy (Bk. IV., Prop. 31) • • (2). 

Multiplying the first equation by x, and the second by a, and sub- 
tracting, member from member, we have 

bx 2 — a 2 b = — ac 2 ; whence 



la (ah - c 2 ) /b (ab - c 2 ) 

*==v — *— ; y ==v «— 

Prop. LXXVIII. — To determine a right-angled triangle, having 
given the lengths of two lines drawn from the vertices of the acute 
angles to the middle points of the opposite sides. 



58 KEY. 

Let ABC be a right angled triangle, and AD, CE, two lines drawn 
to the middle points D and E of the opposite sides. 
Let AD = a 
CE = b 
AB = 2x 
BC = 2y 
Then, 4x 2 + y 2 = a 2 (Bk. IV. Prop. 11) . (l) 
and, 4y 2 -f x 2 — b 2 " " " " (2). 

Multiplying equation (1) by 4, and subtracting (2) from it, we have, 

15z 2 = 4a 2 - b 2 ; when, x = ± x/^ 1 




-b 2 ^ 



and substituting in equation (2) y = ± \ - 



4b 2 — a 2 



15 



Prop. LXXIX. — To determine a right-angled triangle, having given 
the perimeter and the radius of the inscribed circle. 

1st. Let ABC be a right angled triangle, O the centre, and r the 
radius of the inscribed circle. 

Let p = the perimeter ; let x denote the length 
of the equal tangents drawn from A (Bk. III. Prop. 
14, sch.) ; v the length of the equal tangents drawn 
from B ; and y the length of the equal tangents A 
drawn from C. 

Then, AC = x + y, AB = x -f v, and BC = y -f v ; then, 
AC + AB -f BC = 2x + 2y + 2v = p ; 
transposing and reducing, 

x -f y =— - — = a, a known quantity • • • (1). 




PEOBLEMS FEOM LEGENDEE. 59 

Then, AB 2 + BC 2 = AC 2 : that is, 

x 2 + 2vx + v 2 + y 2 + 2vy +v 2 = a 2 • • • • (2). 
2nd. Observe that the double area of each of the triangles AOB, 
BOC, and AOC, is equal to its base multiplied by the radius of the 
inscribed circle ; and hence, the sum of these products is equal to 
the sum of the bases multiplied by r ; that is, = r X p, a known 
quantity. 

But the base AB x BC is also equal to double the area of the 
triangle ABC ; hence, 

(x -h v) x (y + v) = r X p ; that is, 
xy -f vx + vy + v 2 = r X p . . . . (3) ; 
Multiplying both numbers of equation (3) by 2, we have, 
2xy + 2vx + 2vy 4- 2v 2 = 2rp . . . (4) ; 
subtracting equation (4) from (2), we have, 

x 2 — 2xy -\- y 2 = a 2 — 2rp ; whence, by extracting square root, 

x — y = ± -y/a 2 — 2rp = b (5); 

combining (1 ) and (5) we readily find the values of x and y 9 

a -f- b a — b 

* = —£-, and y= .—_ . 

Prop. LXXX. — To determine a triangle, having given the base, 
the perpendicular, and the ratio of the two sides. 

Let ACB be a triangle, and CD perpendicular to the base AB. 
Let AC = y, 

r = ratio ; JP 

then, ry = CB 

AB = b 
DB = b - x 
AD = x 
CD = h. 




60 



KEY. 



Then, y 2 = x 2 + h 2 ; and r 2 y 2 = b 2 — 2bx -f x 2 4- h 2 . 
Multiplying the first equation by r 2 , and subtracting, 

= b 2 — 2bx -{- x 2 -\- h 2 — r 2 x 2 — r 2 h 2 ; whence, 



(1 — r 2 ) 



2b 

x 2 — - • x 

1 — r l 



2bx ■= (r 2 — l)k 2 — b 2 ; or, 

( r 2-\)h 2 -b 2 

= — - — ; whence, 

1 — r z 



x = 



6± -x/l(r 2 - 1) h 2 - b 2 ] (1 - r 2 ) + b 2 



1 — r 2 

Prop. LXXXI. — To determine a right-angled triangle, having 
given the hypothenuse, and the side of the inscribed square. 

Let ACB be a right-angled triangle, and FDEB an inscribed 
square. 

Let AC = h, and DF = s : also, 
denote AB by #, and BC by y\ then CE=y— s. 
Then, AB : BC : : DE : EC ; 
that is, x : y : : s : y — s ; 

whence, xy — sx = sy\ or, 

«y = sy + sz = s (x + y) • • (1) ; 




also, 



4- y 2 = h 2 (Bk. IV., Prop. 11) 



(2). 



If to equation 2, we add twice equation (1), we have 
x 2 4- 2xy 4- y 2 = 2s (x -\- y) -f h 2 ; or, 
(* 4- 2/) 2 - 2s 4- y) = h 2 . . (3), 

which is an equation of the second degree, in which the unknown 

quantity is x 4* y \ hence, 

x 4- y - s 4- V h2 + s * ' ' ( 4 )- 
Combining equations (1) and (4), we have 

xy = s 2 4- * -\/h 2 4- s 2 • • (5). 



PROBLEMS FROM LEGENDRE. 



61 



From the square of equation (4), subtract 4 times equation (5), 
and we have 



x 2 — 2xy + y 2 = h 2 — 2s 2 — 2s i/h 2 + s 2 
extracting the square root of both members, 



x — y =\Jh 2 -2s 2 — 2s ^/h 2 + s 2 
combining equations (4) and (7), we have 



(6); 



P), 



s + -/A 2 + s 2 + \A 2 - 2s 2 — 2s i/h 2 -f s 2 



2 



y = 



s + y/h 2 + s 2 — -y/h 2 — 2s 2 — 2s ^h 2 + s 2 



2 



(8), 



(9). 



Prop. LXXXII. — To determine the radii of three equal circles, 
described within and tangent to, a given circle, and also tangent to each 
other. 

Let O be the centre of the 
given circle, and A, B and C, 
the centres of the equal in- 
scribed circles. 

Denote the radius of the 
given circle by R, and the 
equal radii of the inscribed 
circles by r. 

Joining the centres A and 
C, C and B, B and A, by 
straight lines, we have the 
equilateral triangle ABC, each 
of whose sides is 2r. Draw COD and prolong it to E, ar d it will 

be perpendicular to AB. 

13 




62 KEY. 

Then, in the right-angled triangle ACD, we have 
CD 2 = AC 2 - AD 2 , or 

CD 2 = 4r 2 - r 2 = 3r 2 , or CD = r -y/ST 

But since O is the point at which lines drawn from the vertices of 
the angles to the middle points of the opposite sides, in both tri- 
angles, intersect each other, it follows, from Cor. of Prob. 21, that 

CO = f CD = | r V^; hence, 

OE = R = r + |y-v/3; hence, finally, 



3R 



R 



r = 



3 + 2V§ l+2-v4 

Prop. LXXXIII. — In a right-angled triangle, having given the 
perimeter and the perpendicular let fall from the right angle on the 
hypothenuse, to determine the triangle. 

Let ACB be a right-angled triangle, right angled at C ; and let 
CD be drawn perpendicular to the hypothenuse AB. 

Let p = the perimeter, and 

h == the perpendicular CD. 
Denote AC by x, and CB by y ; 
then, AB = ^/x 2 + y 2 , and A 

AC + CB + AB —p; that is, 

x 4- y + -\A 2 + y 2 = p • (1). 

Again, xy = double the area of ACB • • (2), 

and h y/x 2 -+- y 2 = double area • • (3) ; 

hence, xy — h ^/x 2 -f y 2 • • • • (4) 




PEOBLEMS FEOM LEGEKDKE. 63 

transposing in equation (1) and squaring, 



(x + y) 2 — (p — y/x 2 + y 2 ) 2 , 



x 2 -f- 2xy + y 2 = r p 2 — %P -\A 2 4- y 2 -\- x 2 + y 2 , 



or 2xy = p 2 — %p -y/x 2 4 y 2 • • (5) ; 

multiplying both members of equation (4) by 2, 

2xy — 2A y/x 2 + y 2 . . . (6) ; 
combining equations (5) and (b') 3 

2A y/x 2 4- y 2 = ^? 2 — 2p y^ _j_ ^2 . hence, 

J9 2 



i/s 2 + V 2 = • • • (7) • 



substituting this value of y/x 2 -\- y 2 , in equation (4), 

hp 2 /rt% 

?=.f|fHFrt • • ; (8): 

squaring both members of equation (7), we have 
x 2 4 y 2 — 



4 (A 4- ^) 2 ' 
adding and subtracting 2 times each member of (8), 

* + 2*y + y' = ——-^-Z . . . (9), 



2 o i 2 p 2 (p 2 — 4ph—4h 2 ) 
* - Zxy + y* = a(A +- p y- -* 

Extracting the square roots of equations (9) and (10). 



(10). 



X + 2,_ 2(A+ P ) (H * 



*- y =87*+3 • (13); 



64 KEY. 



p (p + 2h) + ^ yp^-^h-^h 2 
hence, .=-4 ^-j . (13), 

J> (j> + 2ft) -p^/p 2 - 4ph - 4A 2 " 

Prop. LXXXIV. — To determine a right-angled triangle, having 
given the hypothenuse and the difference of two lines drawn from the 
two acute angles to the centre of the inscribed circle. 

Let ABC be a right-angled triangle, right-angled at B, and AO, OC, 
two lines drawn from the acute angles to the centre O of the in- 
scribed circle. 

Let AC = A; AO - OC = d, OC = z; 
then, AO = x -f- d 

Produce AO, and from C draw CD per- 
pendicular to the prolongation, meeting it 
at D. Then, since the sum of the angles 
BAC and ACB is equal to a right-angle 

(Bk. I, Prop. 25, Cor. 4), and since the lines AO and CO bisect 
these angles (Bk. Ill, Prob. 14), OAC + ACO is equal to a half 
a right-angle. 

Since the outward angle COD is equal to the sum of the inward 
angles (Bk. I, Prop. 25, Cor. 6), it is equal to half a right-angle ; 
and hence, OCD is equal to half a right-angle, and hence OD and 
CD are equal. Denote either by z ; then, 

x 2 =. z 2 + z 2 ; and z = x <\f\* 
Then, AC 2 = AD 2 4- CD 2 ; that is, 

k 2 = (x + d+x Vi) 2 4- {x -y/j) 2 ; 

and by reduction, (2 4- Vty &*H= d(2 +^/2)x = h 2 -d 2 ; or, 

h 2 - d 2 

x 2 4- dx = - — 

2+V^ 




PEOBLEMS FKOM LEGENDEE. 



65 



hence, 



*=-J<*+J\/ 



4A» — (2— V§)eP 



2+V^" 



2 2V 2+v^ 

Let OC = x = m, AO = m + c?, CD = z = m^/^, 
AB = y, and CB = w. 
Then, since the triangles ACD and AOG are similar, 

h : m\\ : : m + d : OG = r = — * -; and 

, , /=■ , A ~ (m + c?) (m + ^ + m/i) 
A : m -|- d + my% : : m + d \ AG = — — -- LL 

\B = v ™ AG + GB = ( m +^) ( m + <* + m Vi) + w VT(^ + <0 



A 



tt = -/A 2 - 6 2 . 

Prop. LXXXV. — To determine a triangle, having given the base, 
the perpendicular, and the difference of the two other sides. 

Let ACB be a triangle, and CD perpendicular to the base AB. 
Let AB = o, CD = a, AC - BC = d, 
BC = x, AC = x+d, and DB = z, 
and AD — b—z. 

Then, AC = </«*+(&-«)»; 

and CB = y / a 2 ^fl 2 ; hence, 

y^2 + (6 _ g)8 _ ^/ a 2 _|_ 2 2 _ rf> 

Transposing and squaring, we have 




a 2 + (5 _ ^)2 _ d 2 _j_ 2d -/a 2 + z 2 -f a 2 + * 2 ; 



66 KEY. 

squaring b - z, in the first member, we have 

a 2 + i% _ 2hz + z 2 = d 2 + 2J -/a 2 + z 2 + a 2 + « 2 ; 
w hence, by reducing, 



(62 _ &) - 2bz = 2d -v/a 2 + z 2 ; 
squaring both members, we have 

(62 __ ^2)2 _ 46 (62 _ d 2 ).z + 4b 2 z 2 = 4a 2 d 2 + 4rf 2 z 2 ; 
transposing, and collecting the terms, 

4 (62 _ d 2 ) .z 2 - 46 (b 2 -d 2 ).z = 4a 2 d 2 - (b 2 - d 2 f ; or, 

± a 2 d 2 _ (62 _ ^2)2 

s *-bz = - (fc2 _ d2) ; whence, 



b 1 / 
= 2 ± 2V- 



4a 2 d 2 + 2W — d* 



b 2 -d 2 ' 

from which the sides AC and CB are easily found. 

Prop. LXXXVI. — To determine a triangle, having given the base, 

ike. perpendicular, and the rectangle of the other sides. 

Let ACB be a triangle, and CD a line 
drawn perpendicular to the base AB. 

Let AB = b, CT> = d, AC x CB = q ; 

AC = x, and BC == y ; then, 

xy = q • • . (1). 




== -\/x 2 — d 2 , and DB = i/y 2 — d 2 ; hence, 

y/x 2 - d 2 + ^/y^^d 2 = 6 . . . (2). 

Transposing, ani then squaring both members of equation (2), we 
have 



^ _ d 2 = 6 2 — 2b- x /y 2 —d 2 i- 2J 2 — d 2 ; 



PEOBLEMS FEOM LEGEKDEE. 67 



whence, (x 2 — y 2 ) — b 2 = — 2b -y/y 2 — d 2 ; squaring again, 
x* — 2x 2 y 2 + y* — 2b 2 (x 2 — y 2 ) + 6 4 = 4b 2 y 2 — 4b 2 d 2 ; reducing, 
x 4 — 2x 2 y 2 -f y 4 — 2b 2 (x 2 + y 2 ) = — 6 4 — 46 2 d a • • (3). 
Adding 4 times the square of each member of (1), 

#* + 2x 2 y 2 + y 2 — 2b 2 (x 2 + y 2 ) - 4? 2 — 6 4 — 46 2 d 2 ; or, 
[X 2 + y 2 ) 2 — 26 2 (x 2 + y 2 ) = 4^ 2 — b* — 4b 2 d 2 . 
Regarding x 2 -\- y 2 as a single unknown quantity, we have 



x 2 + y 2 = b 2 db i/4q 2 - 4b 2 d 2 • • • (4) ; 
then, by adding twice each member of equation (1), 



x 2 -f 2xy + x 2 = 2q -f b 2 ± ^/4q 2 - 4b 2 d 2 ; or, 



x + y = \2q + b 2 ± -y/4<7 2 — 46 2 d 2 = m ; 
subtracting from equation (4), twice equation (1), 



x 2 — 2xy-\-x 2 =—2q + b 2 ± -y/4? 2 — 4« 2 ; or, 



n 



x — y = =fc y/— 2q + 6 2 ± V 4 ^ 2 - 462c?2 = 

m -f- w W2. — n 

hence, a; = — - — , and y = — ^ — • 

Prop. LXXXVII. — To determine a triangle, having given the 
lengths of three lines drawn from the three angles to the middle of 
the opposite sides. 

Let ACB be a triangle, and AE, BG, CD, three lines drawn from 
the vertices to the middle points of the opposite sides. 

Let CD=a, AE = 6, BG=c, A.C=x, 
BC=y, and AB=z: then, 

z 2 + y 2 = 2a 2 + j- (Bk. IV. Prop. 14), 



x 2 + z 2 = 2b 2 + ^ ; y 2 +z 2 =2c 2 + ~ ; - 




68 



KEY. 



clearing of fractions and transposing, we have 
2x 2 + 2y 2 -z 2 = 4a 2 •■ • • (1) 



2x 2 +■ 2z 2 - y< 



4b 2 



• • (2) 

2 y 2 + 2z 2 -x 2 -4c 2 . • . (3) ; 
subtracting equation (2) from (1), we have 

multiplying equation (3) by 2, and adding to (2) 

3y 2 + 6z 2 - 8c 2 + 46 2 • 
subtracting equation (4) from (5), 

9z 2 = 8c 2 -f 4b 2 - 4a 2 + 46 2 ; 




z = 


± f y/2b 2 


+ 2c 2 - 


-a 2 


X z= 


±jv^ 


+ 26 2 - 


-c 2 



y = ± J v / 2a^ + 2c 2 — 6 2 . 

Obr. Through G and E, the middle points of AC and BC, draw 
GE ; then will GE be parallel to the base AB (Bk. IV^. Prop. 16) : 
and since AC is equal to twice CG, AD will be equal to twice GP, 
or its equal PE. 

But the triangles AOD and POE are similar ; and, since AD is 
equal to twice PE, AO is equal to twice OE ; that is, 

If three lines be drawn from the vertices of the three angles of a 
triangle to the middle points of the opposite sides, the distance from 
either vertex to the point of intersection, will be two-thirds of the 
bisecting line. 

Prop. LXXXVIII. — In a, triangle, having given the three sides, to 
find the radius of the inscribed circle. 

Let ABC be a triangle, CD perpendicular to the base, O the 
'centre, and OP the radius of the inscribed circle. 



PROBLEMS FROM LEGENDRE. 



69 



Let 



BC = a, AB = c, AC = 6, 

OP = r, AD = z, CD = x. 
x 2 = 62 _ z % . . . (i) y 

# 2 = a 2 — (c — zf ; whence, 
S 2 — z 2 = a 2 — c 2 + 2cz — 2 2 ; therefore, 
b 2 -\- c 2 — a 2 




3> B 



S = — 
equation (1), 


2c 


> 








V^&c 2 


-(b 2 + 


c 2 — 


a 2 )' 


$ 


area = 


(a + b 


2c 
+ c)±r 


— 1 

— 2 


c#; 


whence, 


ex 


\/4W — 


(6 2 


+ 


s 2 — a 2 ) 2 



a -f- 6 + c 



2 (a + 6 + c) 



Prop. LXXXIX. — To determine a right-angled triangle, having 
given the side of the inscribed square, and the radius of the inscribed 
circle. 

Let ABC be a right-angled triangle, with a square and circle both 
inscribed. 

Let AB = x, BC == y, and AC == z: 

denote the side of the square by s, and the radius 
of the circle by r. 

Then, x -f y — z = 2r (Prop. LXXIX, Key) . (1), 



x 2 -f- y 2 = z 2 



(2). A 




and x + y = s + \/z 2 + s 2 (Prop. LXXXI, Key) . . (3). 

Combining equations (1) and (3), and transposing, 

z + 2r — s =z \/z 2 -f- s 2 ; squaring, 
& + ±r 2 + s 2 + ±rz — 2s2 — 4rs = z 2 + s 2 ; 



70 KEY. 

2rs — Zr 2 , x 

whence, z = ^ _ g . . . (4 ) ; 

combining equations (4) and (1), 

2r 2 
a? -f- y = ■=. m, and equation (2) gives 

(2rs — 2r 2 ) 2 . , 

x 2 4- y 2 = -S r^~- = n 2 ; whence, 

^ * (2r — s) 2 

a; = ^ (m -f- V2^ — «% 2 ), 
y = \ (m — ^/2^ 2 — m 2 ). 

Prop. XC. — To determine a right-angled triangle, having given the 
hifpothenuse and radius of the inscribed circle. 

Let ABC be a right-angled triangle, and O the centre of the in- 
scribed circle. 

Let AC — ^ AB = x, BC = y, 

and r = the radius of the circle. 

x + y = & + 2r (Prop. LXXIX, Key) . (1) /£ip 

z 2 + ?/ 2 = h 2 (2). i 

Since the perimeter is equal to 2A -+- 2r, and since four times the 
area is equal to the perimeter into 2r, 

2xy = 4r 2 -f 4hr . . . (3) ; 

subtracting equation (3) from (2), and extracting the square root, 

x — y = 'y/h 2 — 4r 2 — 4rh . . . (4) ; 
combining (4) and (1), 

h -f 2r -f V^ 2 — 4r 2 — 4rA 




^ 



a; — . 



h -f 2r — <\/W— 4r 2 — 4rh 



y = 2 



f.HE JNyVTIOJNyVL jE>EF^IEg Of T^DERg. 



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IPrice 25 Cents. 

This unique, book, published in 1872, is the first to be consistently printed i* 
imitation of writing ; that is, it teaches orthography as we use it. It is for tha 
smallest class cf learners, who soon become familiarized with words by their forms,, 
and learn to read writing while they spell. 

EXTRACT FROM THE PREFACE. 

Sucoess in teaching English orthography is still exceptional, and it mast so con- 
tinue until the principles involved are recognized in practice. Form is foremost : 
the eye and the hand must be trained to the formation of words ; and since spelling 
is a part of writing, the written form only should be used. The laws of mental 
association, also — especially those of resemblance, contrast, and contiguity in time 
and place — should receive such recognition in the construction of the text-book as 
shall insure, whether consciously or not, their appropriate use and legitimate re- 
sults. Hence, the spelling-book, properly arranged, is a necessity from the first ; 
and, though primers, readers, and dictionaries may serve as aids, it can have no 
competent tubstitute. 

Consistently with these views, the words used in the Independent Child's Speller 
have such original classifications and arrangements in columns — in reference to 
location, number of letters, vowel sounds, alphabetic equivalents, and consonant 
terminations— as exhibit most effectively their formation and pronunciation. The 
vocabulary is strictly confined to the simple and significant monosyllables in com- 
mon use. He who has mastered these may easily learn how to spell and pronounce 
words of more than one syllable. 

The introduction is an illustrated alphabet in script, containing twenty-six pic- 
tures of objects, ind their names, commencing both with capitals and small letters. 
Part First embraces the words of one, two, and three letters; Part Second, the 
words of four letters ; and Part Third, other monosyllables. They are divided into 
short lists and arranged in columns, the vowels usually in line, so as to exhibit in- 
dividual characteristics and similarity of formation. The division of words into 
paragraphs is shown by figures in the columns. Each list is immediately followed 
by sentences for reading and writing, in which the same words are again presented 
with irregularities of form and sound. Association is thus employed, memory 
tested, and definition most satisfactorily taught. 

Among the novel and valuable features of the !essor,s and exercises, probably the 
most prominent are their adaptedness for young- -ohildren and their being printed in 
exact imitation of writing. The author believes that hands large enough to spin a 
top, drive a hoop, or catch a ball, are not too small to use a crayon, or a slate and 
pencil; that the child's natural desirfe to dvaw and write fciould not be thwarted, 
but gratified, encouraged," and wisely directed ; and that since the written form is 
the one actually used in connection with spelling in after-life, the eye and the hand 
of the child should be trained to tnat form from the first. He hopes that this little 
work, designed to precede all other spelling-books and conflict with none, ma? 
satisfy the need so universally recognized Of a fit introduction to orthography, per* 
clanship, and Eng^sh composition. 

5 



The National Series of Standard Sc?iool-!Boofcs. 



The National Readers and Spellers. 

THEIR RECORD. 

These "books have been adopted by the School Boards, or official authority, of 
the following important States, cities, and towns— in most cases for exclusive use, 

The State of Minnesota, The State of Texas. 

The State of Missouri. 

The State of Alabama. 

The State of North Carolina. 
The State of Louisiana. 

Indiana. 
New Albany. 
Fort Wayne. 
Lafayette. 
Madison. 
Logansport. 
Indianapolis. 

Iowa. 

Davenport. 
Burlington. 
Muscatine. 
Mount Pleasant. 
&c. 

Nebraska. 
Brownsville. 
Lincoln. 

&c. 

Oregon. 

Portland. 
Salem. 
&c. 

Virginia. 
Richmond. 

Norfolk. 
Petersburg. 
Lynchburg. 
&c. 

South Caroling 

Columbia. 

Charleston. 

Georgia. 
Savannah. 

Louisiana. 
New Orleans. 

Tennessee 

Memphis 

$he Educational Bulletin records periodically all new points gained 



ISTew York. 


Illinois. 


New York City. 


Chicago. 


Brooklyn. 


Peoria. 


Buffalo. 


Alton. 


Albany. 


Springfield. 


Rochester. 


Aurora. 


Troy. 


Gales burg. 


Syracuse. 


Rockford. 


Ehnira. 


Rock Island. 


&c, &c. 


&c, &c. 


Pennsylvania, 


"Wisconsin, 


Reading. 


Milwaukee. 


Lancaster. 


Fond du Lac. 


Erie. 


Oshkosh. 


Scran ton. 


Janesville. 


Carlisle. 


Racine. 


Carbondalo. 


Watertown. 


Westchester. 


Sheboygan. 


Schuylkill Haven, 


La Crosse. 


Williamsport. 


Waukesha. 


Norristown. 


Kenosha. 


Bellefonte. 


&c, cfec. 


Wilkesbarre. 




&c, &c. 






Michigan. 


New Jersey. 


Grand Rapids. 


Newark. 


Kalamazoo. 


Jersey City. 


Adrian. 


Paterson. 


dackson. 


Trenton. 


MoLroe. 


Camden. 


Lansing. 


Elizabeth. 


&c, &c. 


New Brunswick. 




Phillip sburg. 
Orange. 


Ohio. 


&c, &c. 


Toledo. 




Sandusky. 


Delaware. 


Conneaut. 


Wilmington. 


Chardon. 
Hudson. 


D. C. 


Canton. 
Salem. 


Washington. 


&c, &c. 



The National Series of Standard School-Hooks. 



SCHOOL-ROOM CARDS. 

* 9 ♦ 

Baade's Reading Case, **io oo 

A frame containing movable cards, with arrangement for showing 
one sentence at a time, capable of 28,000 transpositions. 

Eureka Alphabet Tablet *i 50 

Presents the alphabet upon the Word Method System, by which the 
child will learn the alphabet ia nine days, and make no smail progress in 
reading and spelling in the same time. 

National School Tablets, 10 Nos *8 oo 

Embrace reading and conversational exercises, object and moral les- 
sons, form, color, &c. A complete set of these large and elegantly illus- 
trated Cards will embellish the school-room more than any other article 
of furniture. 

READING. 



» » » 



Fowle's Bible Reader $100 

The narrative portions of the Bible, chronologically and topically ar- 
ranged, judiciously combined with selections from the Psalms, Proverbs, 
and other portions which inculcate important moral lessons or the great 
truths of Christianity. The embarrassment and difficulty of reading the 
Bible itself, by course, as a class exercise, are obviated, and its use made 
feasible, by this means. 

North Carolina First Reader 40 

North Carolina Second Reader 65 

North Carolina Third Reader 3 00 

Prepared expressly for the schools of this State, by C. R*. "Wiley, Super- 
intendent of Common Schools, and F. M. Hubbard, Professor of Litera- 
ature in the State University. 

Parker's Rhetorical Reader 1 00 

Designed to familiarize Readers with the pauses and other marks in 
general use, and lead thean to the practice of modulation and inflection of 
the voice. 

Introductory Lessons in Reading and Elo- 
cution 75 

Of similar character to the foregoing, for less advanced classes. 

High School Literature 3 50 

Admirable selections from along list of the world's best writers, for ex- 
ercise in reading, oratory, and composition. Speeches, dialogues, and 
model letters represent the latter department, 

7 



The National Series of Standard School-ljOGks* 

ort hograp hy" 

SMITH'S SERIES ; 

Supplies a speller for every class in graded schools, and comprises the most com- 
plete and excellent treatise on English Orthography and its companion 
branches extant. 

1. Smith's Little Speller % m 

First Round in the Ladder of Learning. 

2. Smith's Juvenile Definer 45 

Lessons composed of familiar words grouped with reference to similar 
signification or use, and correctly spelled, accented, and defined. 

3. Smith's Grammar-School Speller .... 50 

Familiar words, grouped with reference to the sameness of sound of syl- 
lables differently spelled. Also definitions, complete rules for spelling and 
formation of derivatives, and exercises in false orthography. 

4 Smith's Speller and Definer's Manual . so 

A complete School Dictionary containing 14,000 words, with various 
other useful matter in the way of Rules and Exercises. 

5. Smith's Etymology— Small, 75; Complete . 1 25 

The first and only Etymology to recognize the Anglo-Saxon our mother 
tongue; containing also full lists of derivatives from the Latin, Greek, 
Gaelic, Swedish, Norman, &c, &c ; being, in fact, a complete etymology 
of the language for schools. 

Sherwood's Writing Speller 15 

Sherwood's Speller and Definer 15 

Sherwood's Speller and Pronouncer ... 15 

The Writing Speller consists of properly ruled and numbered blanks 
to receive the words dictated by the teacher, with space for remarks and 
corrections. The other volumes may be used for the dictation or ordinary 
class exercises. 

Price's English Speller . *i5 

A complete spelling-book for all grades, containing more matter than 
"Webster," manufactured in superior style, and sold at a lower price — 
consequently the cheapest speller extant. 

Northend's Dictation Exercises 63 

Embracing valuable information on a thousand topics, communicated 
in such a manner as at once to relieve the exercise of spelling of its usual 
tedium, and combine it with instruction of a general character calculated 
to profit and amuse. 

Wright's Analytical Orthography .... 25 

This standard work is popular, because it teaches the elementary sounds 
In a plain and philosophical manner, and presents orthography and or- 
thoepy in an easy, uniform system of analysis or parsing. 

Fowle's False Orthography ....... 45 

Exercises for correction. - 

Page's Normal Chart- ... *3 75 

The elementary sounds of the language for the school-room walla. 

8 



The JYatio?ial Series of Standard School-Books. 

ORTHOGRAPHY-Continued. 

Barber's Critical Writing Speller 20ets. 

" The Student's Own Hand-Book of Orthography, Definitions, and Sentences, 
consisting of Written Exercises in the Proper Spelling, Meaning, and Use of 
Words." (Published 1873.) This differs from Sherwood's and other Writing 
Spellers in its more comprehensive character. Its blanks are adapted to writing 
whole sentences instead of detached words, with the proper divisions for number- 
ing, corrections, etc. Such aids as this, like Watson's Child's Speller and Sher- 
wood's Writing Speller, find their raison d'etre in the postulate that the art of cor- 
rect spelling is dependent upon written, and not upon spoken language, for its util- 
ity, if not for its very existence. Hence the indirectness of purely oral instruction, 



ETYMOLOGY. 

Smith's Complete Etymology, $1 25 

Smith's Condensed Etymology 75 

Containing the Anglo-Saxon, French, Dutch, German, Welsh, Danish, Gothic, 
Swedish, Gaelic, Italian, Latin, and Greek Roots, and the English words derived 
therefrom accurately spelled, accented, and defined. 



From Hon. Jno. G. McMtot, late State Superintendent of Wisconsin. 
I wish every teacher in the country had a copy of this work. 

From Pein. War. F. Phelps, Minn. State Normal. 

The book is superb— just what is needed in the department of etymology and 
spelling. 

From Prof. C. H. Verrile, Pa. State Normal School. 

The Etymology (Smith's) which we procured of you we like much. It is the 
best work for the class-room we have seen. 

From Hon. Edward Baeeard, Supt. of Common Schools, State of Maine. 

Many a teacher who has turned his attention to the derivation of words has 
rejoiced in the helps furnished by dictionaries and smaller "hand-books," where 
his taste could be gratified, and the labors of patient students have been available 
to his own improvement. A treatise on this subject, called U A Complete Ety- 
mology of the English Language," contains very much information in a small space. 
The author, W. W. Smith, is evidently a lover of this branch of study, and has fur' 
nished a manual of singular utility for its purpose. 



DICTIONARY. 



"C^^" 



The Topical Lexicon, . . ; . 175 

This work is a School Dictionary, an Etymology, a compilation of synonyms, and 
a manual of general information. It differs from the ordinary lexicon "in being 
arranged by topics instead of the letters of the alphabet, thus realizing the apparent 
paradox of a "Readable Dictionary." An unusually valuable school-book. 

9 



The National Se?'/es of Standard School-Books, 



CLARK'S DIAGRAM SYSTEM, 



Clark's Easy Lessons in Language, . . . $o 35 

Published 1874. Contains illustrated object-lessons of the most attractive charac. 
tev, and is couched in language freed as much as possible from the dry technicalities 
of the science. 

Clark's Brief English Grammar, 60 

Published 1872. Part I. is adapted to youngest learners, and the whole forms a 
complete ■' brief course " in one volume, adequate to the wants of the common 
school. 

Clark's Normal Grammar, 1 oo 

Published 1870, and designed to take the place of Prof. Clark's veteran "Prac- 
tical" Grammar, though the latter is still furnished upon order. The Normal is 
an entirely new treatise. It is a full exposition of the system as described below, 
with all the most recent improvements. Some of its peculiarities are — A happy 
blending of Syntheses with Analyses ; thorough Criticisms of common errors 
in the use of our Language ; and important improvements in the Syntax of Sen- 
tences and of Phrases. 

Clark's Key to the Diagrams, .».,.'. l 6b 
Clark's Analysis of the English Language, . 60 
Clark's Grammatical Chart, *3 75 

The theory and practice of teaching grammar in American schools is meeting 
with a thorough revolution from the use of this system. While the old methods 
offer proficiency to the pupil only after much weary plodding and dull memorizing, 
this affords from the inception the advantage of practical Object Teaching, address- 
ing the eye by means of illustrative figures ; furnishes association to the memory, 
its' most powerful aid, and diverts the pupil by taxing his ingenuity. Teachers 
who are using Clark's Grammar uniformly testify that they and their pupils find it 
the most interesting study of the school course. 

Like all great and radical improvements, the system naturally met at first with 
much unreasonable opposition. It has not only outlived the greater part of this 
'opposition, but finds many of its warmest admirers among those who could not 
at first tolerate so radical an innovation. All it wants is an impartial trial to con- 
vince the most skeptical of its merit. No one who has fairly and intelligently 
tested it in the school-room has ever been known to go back to the old method. 
A great success is already established, and it is easy to prophecy that the day is 
not far distant when it will be the only system of teaching English Grammar. As 
the System is copyrighted, no other text-books can appropriate this obvious an£ 
great improvement. 



-<v 



Welch's Analysis of the English Sentence, • i 25 

Remarkable for its new and simple classification, its method of treating connec- 
tives, its explanations of the idioms and constructive laws of the language, etc. 

10 



The National Series of Standard School-Hooks. 

Clark's Diagram English Grammar. 

TESTIMONIALS. 
From J. A. T. Duenin, Principal Dubuaue R. C. Academy, Iowa. 
In my opinion, it is well calculated by its system of analysis to develop those rational 
faculties which in the old systems were rather left to develop themselves, while the 
memory was overtaxed, and the pupils discouraged. 

« From B. A. Cox, School Commissioner, Warren County, Illinois. 

I have examined 150 teachers in the last year, and those having studied or taught 
•Clark's System have uni/ersally stood fifty per cent better examinations than those 
having studied other authors. 

■ From M. H. B. Bucket, Principal Masonic Institute, Georgetown, Tennessee. 
; I traveled two years amusing myself in instructing (exclusively) Grammar classes 
with Clark's system. The first class I instructed fifty days, but fouud that this was 
more time than was required to impart a theoretical knowledge of the science. 
During the two years thereafter I instructed classes only thirty days each. Invariably 
I proposed that unless 1 prepared my classes for a more thorough, minute, and accu= 
rate knowledge of English Grammar than that obtained from the ordinary books and 
in the ordinary way in from one to two years, I would make no charge. 1 never 
failed in a solitary case to far exceed the hopes of my classes, and made money and 
character rapidly as an instructor. 

From A. B. Douglass, School Commissioner, Delaware County, New York. 

I have never known a class pursue the study of it under a live teacher, that has not 
succeeded ; I have never known it to have an opponent in an educated teacher who 
had thoroughly investigated it; I have never known an ignorant teacher to examine 
it ; I have never known a teacher who has used it, to try any other. 

From J. A. Dodge, Teacher and Lecturer on English Grammar, Kentucky. 

We are tempted to assert that it foretells the dawn of a brighter age to our mother- 
tongue. Both pupil and teacher can fare sumptuously upon its contents, however 
highly they may have prized the manuals into which they may have been initiated, 
and by which their expressions have been moulded. 

Front W. T. Chapman, Superintendent Public Schools, Wellington, Ohio. 

I regard Clark's System of Grammar the best published. For teaching the analysis 
of the Engliph Language, it surpasses any I ever used. 

From F. S. Lyon, Principal South Norwalk Union School, Connecticut. 

During ten years' experience in teaching, I have used six different authors on the 
Subject of English Grammar. I am fully convinced that Clark's Grammar is better 
calculated to make thorough grammarians than any other that I have seen. 

From Catalogue op Rohbeb's Commeecial College, St. Louis, Missouri. 

We do not hesitate to assert, without fear of successful contradiction, that a better 
knowledge, of the English language can be obtained by this system in six weeks than 
by the old methods in as many months. 

From A. Pickett, President of the State Teachers' Association, Wisconsin. 

A thorough experiment in the use of many approved authors upon the subject of 
English Grammar has convinced me of she superiority of Clark. When the pupil has 
completed the course, he is left upon a foundation of principle, and not upon the die* 
**m of the author. 

From Geo. F. McFaeland, Prin. McAllisterville Academy, Juniata Co., Penn. 

At the first examination of public-school teachers by the county superintendent, 
when one of our student teachers commenced analyzing a sentence according to Clark, 
the superintendent listened in mute astonishmeuc until he had finished, then asked 
what that meant, and finally, with a very knowing look, said Euch work wouldn't do 
here, and asked the applicant to parse the sentence right, and gave the lowest certifi- 
cates to all who barely mentioned Clark. Afterwards, I presented him with a copy, 
and the next fall he permil ted it to be partially used, while the third o' last fall, he 
openly commended the system, and appointed three of my best teachcrr to explain it 
at the two Institutes and one County Convention held since September. 



For further testimony of equal force, see the Publishers' Specia> Circular, el 
current numbers of the Educational Bulletin. 

11 



The National Series of Standard School-3>ooks. 



GRAPHY. 



■ < ■ • & m »»■ 



NATIONAL GEOGRAPHICAL SYSTEM. 



THE SERIES. 

I. Monteith's First Lessons in Geography, . . .. $ C5 

II. Monteith's New Manual of Geography, . . . 1 10 1 

II. McNaUy's System of Geography, 2 00 

INTERMEDIATE OB ALTERNATE VOLUMES. 

!*. Monteith's Introduction to Geography, 63 

2*. Monteith's Physical and Political Geography, ... 1 88 



ACCESSORIES. 
Monteith's Wall Maps 2 sets (see page 15), $-20 03 and *35 00 
Monteith's Manual of Map-Drawing (Allen's System) . 25 
Monteith's Map-Drawing and Object-Lessons, . . 75 
Monteith's Map-Drawing Scale, *25 

1. PRACTICAL OBJECT TEACHING. The infant scholar is first introduced 
to a picture whence he may derive notions of the shape of the earth, the phenom- 
ena of day and night, the distribution of land and water, and the great natural 
divisions, which mere words would fail entirely to convey to the untutored mind. 
Other pictures follow on the same plan, and the child's mind is called upon to grasp 
no idea without the aid of a pictorial illustration. Carried on to the higher 
books, this system culminates in Physical Geography, where such matters as 
climates, ocean currents, the winds, peculiarities of the earth's crust, clouds and 
rain, are pictorially explained and rendered apparent to the most obtuse. The 
illustrations used for this purpose belong to the highest grade of art. 

2. CLEAR, BEAUTIFUL, AND CORRECT MAPS. In the lower numbers the 
maps avoid unnecessary detail, while respectively progressive, and affording the 
pupil new matter for acquisition each time he approaches in the constantly en- 
larging circle the point of coincidence with previous lessons in the more ele- 
mentary books. In the Physical and Political Geography the maps embrace many 
new and striking features. One of the most effective of these is the new plan for 
displaying on each map the relative sizes of countries not represented, thus obvi- 
ating much confusion which has arisen from the necessity of presenting maps in 
tne same atlas drawn on different scales. The maps of " McNally" have long been 
celebrated for their superior beauty and completeness. This is the only school- 
book in which the attempt to make a complete atlas also clear and distinct, has 
been successful. The map coloring throughout the series is also noticeable. 
Delicate and subdued tints take the place of the startling glare of inharmonious 
colors which too frequently in such treatises dazzle the eyes, distract the atten- 
tion, and serve to overwhelm the names of towns and the natural features of tht 
landscape. 

12 



The National Series of Standard School-!Books. 

GEOGRAPHY-Continued. 

3, THE VAEIETT OF MAP EXERCISE, Starting each time from a different 
basis, the pupil in many instances approaches the same fact no less than six 
times^ thus indelibly impressing it upon his memory. At the same time this sys- - 
tern is not allowed to become wearisome — the extent of exercise on each subject 
being graduated by its relative importance or difficulty of acquisition. 

4, THE CHARACTER AND ARRANGEMENT OF THE DESCRIPTIVE 
aEXT. The cream of the science has been carefully culled, unimportant matter 
rejected, elaboration avoided, and a brief and concise manner of presentation cul- 
tivated. The orderly consideration of topics has contributed greatly to simplicity. 
Due attention is paid to the facts in history and astronomy which are inseparably 
connected with, and important to the proper understanding of geography— and 
such only are admitted on any terms. In a word, the National System teaches 
geography as a science, pure, simple, and exhaustive. 

5, ALWAYS HP TO THE TIMES. The authors of these books, editorially 
speaking, never sleep. ISTo change occurs in the boundaries of countries, or of 
counties, no new discovery is made, or railroad built, that is not at once noted 
and recorded, and the next edition of each volume carries to every school-room 
the new order of things. 

6= SUPERIOR G-RADATION. This is the only series which furnishes an avail- 
able volume for every possible class in graded schools. It is not contemplated 
that a pupil must necessarily go through every volume in succession to attain 
proficiency. On the contrary, two will suffice, but three are advised ; and if the 
course will admit, the whole series should be pursued. At all events, the books 
are at hand for selection, and every teacher, of every grade, can find among them 
one exactly suited to his class. The best combination for those who wish tc 
abrklge the course consists of Nos. 1, 2, and 3, or where children are somewhat ad- 
vanced in other studies when they commence geography, Nos. 1*, 2, and 3. "Where 
but two books are admissible, Nos. I*and2*, orlsTos. 2 and 3, are recommended. 

7. FORM OF THE VOLUMES AND MECHANICAL EXECUTION. The maps 

<ind text are no longer unnaturally divorced in accordance with the time-honored 
practice of making text-books on this subject as inconvenient and expensive as 
v ossible. On the contrary, all map questions are to be found on the page opposite 
.ue map itself, and each book is complete in one volume. The mechanical execu- 
tion is unrivalled. Paper and printing are eveiything that could he desired, and 
the binding is— A. S. Barnes and Company's. 

8. MAP-DRAWING-. In 1869 the system of Map-Drawing devised by Profess- or 
Jekome Allex was secured exclusively for this series. It derives its claim to 
originality and usefulness from the introduction of a fixed unit of measurement 
applicable to every Map. The principles being so few, simple and comprehensive, 
the subject of Map-Drawing is relieved of all practical difficulty. (Tn Ncs. 2, 2% 
and 3, and published separately.) 

8 . AN A.LOCOUS OUTLINES. At the same time with Map-Drawing was also in- 
trodr.ced (in No. 2), a new and ingenious variety of Object Lessons, consisting of s 
co-^pariew of the outlines of countries with familiar objects pictorially represented. 

13 



The JSTationag Series of Standard School-'Sooks. 

GEOGRAPHY-Continued. 

MONTEITH'S INDEPENDENT COURSE. 
Elementary Geography (published 1874) . . $o 80 
Comprehensive Geography (with 103 Maps) . 1 60 

^W" These volumes are not revisions of old works— not an addition to any 
series— but are entirely new productions— each, by itself complete, independent, 
comprehensive, yet simple, brief, cheap, and popular ; or, taken together, the most 
admirable " series " ever offered for a common-school course. They present the 
following features, skillfully interwoven— the student learning all about one country 
at a time. 

LOCAL GEOGRAPH1T, or the Use of Maps. Important features of 
the Maps are the coloring of States as objects, and the ingenious system for laying 
down a much larger number of names for reference than are found on any other 
Maps of same size^— and without crowding. 

PHYSICAL GE®GRAPH?T, or the Natural Features of the Earth, 
illustrated by the original and striking Relief Maps, being bird's-eye views or 
photographic pictures of the Earth's surface. 

DESCRIPTIVE GEOGRAPHY, including the Physical ; with some 
account of Governments, and Races, Animals, etc. 

HISTORICAL GEOGRAFHIT, or a brief summary of the salient 
points of history, explaining the present distribution of nations, origin of geo- 
graphical names, etc. 

MATHEMATICAL GEOGRAPHY, including Astronomical.. 
which describes the Earth's position and character among planets ; also the Zones, 
Parallels, etc. 

COMPARATIVE GEOGRAPHY, or a system of analogy, con- 
necting new lessons with the previous ones. Comparative sizes and latitudes arc 
shown on the margin of each Map, and all countries are measured in the '•'•frame 
of Kansas.''' 1 

TOPICAL GEOGRAPHY; consisting of questions for review, and 
testing the student's general and specific knowledge of the subject, with sugges- 
tions for Geographical Compositions. 

ANCIENT GEOGRAPHY*. A section devoted to this subject, with 
Maps, will be appreciated by teachers. It is seldom' taught in our common schools, 
because it has heretofore required the purchase of a separate book. 

GRAPHIC GEOGRAPHY, or Map-Drawing by Allen's "Unit o/ 
Measurement" system (now almost universally recognized as without a rival) 14 
introduced throughout the lessons, and not as an appendix. 

CONSTRUCTIVE GEOGRAPH1T, or Globe-Making. With each 
book a set of Map Segments is furnished, with which each student may make his 
own Globe by following the directions given. 

RAILROAD GEOGRAPH*Sr 7 with a grand Map illustrating routes 
of travel in the United States. Also, a " Tour in Europe." 

14 



The J\Tatio?ial Series 0/ Sta?idard School-Books. 



The National System of Geography, 

By Monteith & McNally. 
its record. 

These popular text-books have been adopted, by official authority, for the 
schools of the following States and Cities — in most cases for exclusive and 
uniform use. 

STATES. 



California, 


Vermont, 


Florida, 




Missouri, 


Iowa, 


Minnesota, 


Alabama, 


Louisiana, 


Nor tli Carolina, 


Tennessee, 


Oregon, 


Kansas, 




Texas, 


Arkansas, 


Mississippi. 




CITIES. 




New York City, 


Louisville, 


Nashville, 


Portland, 


Brooklyn, 


Newark, 


Utica, 


Savannah, 


New Orleans, 


Milwaukee, 


"Wilmington, 


Indianapolis, 


Buffalo, 


Charleston, 


Trenton, 


Springfield, 


Richmond, 


Rochester, 


Norfolk, 


Wheeling, 


Jersey City, 


Mobile, 


Norwich, 


Toledo, 


Hartford, 


Syracuse, 


Lockport, 


Bridgeport, 


Worcester, 


Memphis, 


Dubuque, 


St: Paul, 


San Francisco, 


Salt Lake City, 


Galveston, 


Yicksburg, 


&c. 


&c. 


&c. 


&c. 



STANDARD WALL MAPS. 

By Jambs Monteith. 
Monteith'S School Maps, 8 Numbers, per set *|20 00 

The " School Series " includes the Hemispheres (2 Maps), United States, North 
America, South America, Europe, Asia, Africa. — Price. $2.50 each. 

Each Map is 28 x 34 inches, beautifully colored, has the names all laid down, and 
is substantially mounted on canvas with rollers. 

Monteith'S Grand Maps, 7 Numbers, per set 

(iii locked box) *35 00 

The " Grand Series " includes the Hemispheres (1 Map), United States, South 
America, Europe, Asia, Africa, The World on Mercator's Projection.— Price, $5.00 
each. Size 42 x 52 inches,, names laid down, colored, mounted, &c. like the School 
Series, 

15 



The National Series of Standard Sc/iool-Soo/cs. 

Monteith. & McNally's National Geographies. 

CRITICAL OPINIONS. 

From R. A- Adatos, Member of Board of Education, New York. 

I have found, by examination of the Book of Supply of our Board, that consid' 
erably the largest number of any series now used in our public schools is th<? 
National, by Monteith and McNally. 

From Bro. Patrick, Chief Provincial of the Vast Educational Society of the 
Christian Brothers in the United States. 

Having been convinced for some time past that the series of Geographies i:? 
tise in our schools were not giving satisfaction, and came far short of meeting 
our most reasonable expectations, I have felt it my imperative duty to examine 
into this matter, and see if a remedy could not be found. 

Copies of the different Geographies published in this country have been placed 
at our command for examination. On account of other pressing duties we bav3 
not been able to give as much time to the investigation of all these different sericj 
as we could have desired ; yet we have found enough to convince us that there arc 
mauy others better than those we are now using; but we cheerfully give our most 
decided preference, above all others, to the National Series, by Monteith & McNally. 

Their easy gradation, their thoroughly practical and independent character, 
their comprehensive completeness as a full and accurate system, the wise dis- 
crimination shown in the selection of the subject matter, the beautiful and copious 
illustrations, the neat cut type, the general execution of the works, and other ex^ 
cellencies, will commend them to the friends of education everywhere. 

From the "Home Monthly," NashvilU, Tenn. 
Monteith's and McNally's Geographies. — Geography is so closely con- 
nected with Astronomy, History, Ethnology, and Geology, that it is difficult to 
define its limits in the construction of a text-book. If the author confines himself 
strictly to a description of the earth's surface, his book will be dry, meager, and 
unintelligible to a child. If, on the other hand, he attempts to give information 
©n the cognate sciences, he enters a boundless field, and may wander too far. It 
eeems to us that the authors of the series before us have hit on the happy medium 
between too much and too little. The First Lessons, by applying the system of 
object-teaching, renders the subject so attractive that a child, just able to read, 
may become deeply interested in it. The second book of the course enlarges the 
view, but still keeps to tho maps and simple descriptions. Then, in the third 
book, we have Geography combined with History and Astronomy. A general 
view of tiie solar system is presented, so that the pupil may understand the 
earth's position on the map of the heavens. The first part of the fourth book 
treats of Physical Geography, and contains a vast amount of knowledge com- 
pressed into a small space. It is made bright and attractive by beautiful pictures 
and suggestive illustrations, on the principle of object-teaching. The maps in 
the second part of this volume are remarkably clear, and the map exercises are 
copious and judicious. In the fifth and last volume of the series, the whole sub- 
ject is reviewed and systematized. This is strictly a Geography. Its maps are 
beautifully engraved and clearly printed. The map exercises are full and com- 
prehensive. In all these books the maps, questions and descriptions are given in 
the same volume. Jji most geographies there are too many details and minute 
descriptions— more than any child out of purgatory ought to be required to learn. 
The power of memory is overstrained ; there is confusion — no clearly defined idea 
is formed in the child's mind. But in these books, in brief, pointed descriptions, 
and constant «se of bright, accurate maps, the whole subject is photographed otf 
the mind. 1Q 



T7ie National Series of Standard School-Sooks. 



MATHEMATICS. 



DAVIES' NATIONAL COURSE 



ARITHMETIC, 

. , ~ . SLATED. 

I. Davies Primary Arithmetic, $ 25 $ 32 

2. Davies' Intellectual Arithmetic, 40 48 

3. Davies' Elements of Written Arithmetic, . . , . 50 60 

4. Davies' Practical Arithmetic, 90 1 00 

Key to Practical Arithmetic, 90 

5. Davies' University Arithmetic, 1 40 1 50 

Key to University Arithmetic, *1 40 

ALGEBRA. 

I. Davies' New Elementary Algebra, *1 25 1 35 

Key to Elementary Algebra, *1 25 

2. Davies' University Algebra, 1 50 1 60 

Key to University Algebra, *1 50 

3. Davies' New Bourdon's Algebra, 2 25 2 38 

Key to Bourdon's Algebra, *2 25 

GEOMETRY. 

1. Davies' Elementary Geometry and Trigonometry, 1 40 1 50 

2. Davies' Legendre's Geometry, 2 25 2 38 

3. Davies' Analytical Geometry and Calculus, ... 2 50 2 63 

4. Davies' Descriptive Geometry, 2 75 2 88 

5. Davies' New Calculus, 2 00 

MENSURATION. 

I. Davies' Practical Mathematics and Mensuration, . 1 50 1 60 

2. Davies' Elements of Surveying, 2 50 2 63 

3, Davies' Shades, Shadows, and Perspective,. . . 3 75 3 88 

MATHEMATICAL SCIENCE. 

Davies' Grammar of Arithmetic, * 50 

Davies' Outlines of Mathematical Science, *1 00 

Davies' Nature and Utility of Mathematics, 8vo, *2 00, 12mo, *1 50 

Davies' Metric System, *1 50 

Davies & Peck's Dictionary of Mathematics, *5 00 

Davies' Foundations Mathematical Science, * 25 



t a 

XI 



The National Series of Standard School- Hooks* 

* . ~ . . — , , _ 

MATHEMATICS-Continued. 

ARITHMETICAL EXAMPLES. 

Reuck's Examples in Denominate Numbers % so 
Reuck's Examples in Arithmetic i oo 

These volumes diffel from the ordinary arithmetic in their peculiar^ 
practical character. They are composed mainly of examples, and aaord 
the most severe and thorough discipline for the mind. While a book 
which should contain a complete treatise of theory and practice would be 
too cumbersome for every-day use, the insufficiency of practical examples 
has been a source of complaint. 

HIGHER MATHEMATICS. 

Church's Elements of Calculus 2 50 

Church's Analytical Geometry 2 50 

Church's Descriptive Geometry, with Shades, 

Shadows, and Perspective 4 00 

These volumes constitute the " West Point Course" in their several 
departments. 

Courtenay's Elements of Calculus . . . . 3 oo 

A werk especially popular at the South. 

Hackley's Trigonometry 2 50 

With applications to navigation and surveying, nautical and practical 
geometry and geodesy. 

Peck's Analytical Geometry l 75 

Peck's Practical Calculus-" ....... l 75 

APPLIED MATHEMATICS. 

Peck's Ganot's Popular Physics 1 75 

Peck's Elements of Mechanics 2 oo 

Peck's Practical Calculus 1 75 

Peck's Analytical Geometry, i ? 5 

Prof W G Peck, of Columbia College, has designed the first of these works for 
the ordinary wants of schools in the department of Natural Philosophy. The 
other volumes are the briefest treatises on those subjects now published Iheir 
methods are purely practical, and unembarrassed by the details which rather con- 
fuse than simplify science. 

SLATED ARITHMETICS. 

This consists of the application of an artificially slated surface to the inner cover 
of a book, with flap of the same opening outward, so that students may refer to 
the book and use the slate at one and the same time, and as though the slate were 
detached. When folded up, the elate preserves examples and memoranda tiL 
needed. The material used is as durable as the stone slate. The additional cost 
•jf books thus improved is trifling. 



JVatio?ial Series of Standard School-Hoofcs. 

Davies' National Course of Mathematics. 

TESTIMONIALS. 

From L. Van Bokkelen, State Superintendent Public Instruction, Maryland. 

The series of Arithmetics edited by Prof. Davies. and published by your firm, 
aave been used for many years in the schools of several counties, and the city oi 
Baltimore, and have been "approved by teachers and commissioners. 

Under the law of 1865, establishing a uniform system of Free Public Schools, 
these Arithmetics, were unanimously adopted by the State Board of Education, 
after a careful examination, and are now used in all the Public Schools of Mary- 
: |and. 

These facts evidence the high opinion entertained by the School Authorities ol 
the value of the series theoretically and practically. 

From Horace Webster, President of the College of New York. 
The undersigned has examined, with care and thought, several volumes of Da- 
vies' Mathematics, and is of the opinion that, as a whole, it is the most complete 
and best course for Academic and Collegiate instruction, with which he is ac- 
quainted. 

From David N. Camp, State Superintendent of Common Schools, Connecticut. 

I have examined Da vies' Series of Arithmetics with some care. The language 
is clear and precise ; each principle is thoroughly analyzed, and the whole so ar 
ranged as to facilitate the work of instruction. Having observed the satisfaction 
and success with which the different books have been used by eminent teachers, 
it gives me pleasure to commend them to others. 

From J. O. "Wilson, CJiairman Committee on Text-Boohs, Washington, D. C. 

I consider Davies' Arithmetics decidedly superior to any other series, and in 
this opinion I am sustained, I believe, by the entire Board of Education and Corps 
of Teachers in this city, where they have been used for several years past. 

From John L. Campbell, Professor of Mathematics, Wabash College, Indiana. 
A proper combination of abstract reasoning and practical illustration is the 
chief excellence in Prof. Davies' Mathematical works. I prefer his Arithmetics, 
Algebras, Geometry and Trigonometry to all others now in use, and cordially re- 
commend them to all who desire the advancement of sound learning. 

From Major J. H. Whittlesey, Government Inspector of Military Schools. 

Be assured, I regard the works of Prof. Davies, with which I am acquainted, as 
by far the best text-books in print on the subjects which they treat. I shall cer- 
tainly encourage their adoption wherever a word from me may be of any avail. 

From T. McC. Ballantlne, Prcf. Mathematics Cumberland College, Kentucky. 
\ have long taught Prof. Davies' Course of Mathematics, and I continue to like 
cheir working. 

From John McLean Bell, B. A., Prin. of Lower Canada College. 

I have used Davies' Arithmetical and Mathematical Series as text-books in the 
schools under my charge for the last six years. These I have found of great effi- 
cacy in exciting, invigorating, and concentrating the intellectual faculties of the 
young. 

Each treatise serves as an introduction to the next higher, by the similarity of 
its reasonings and methods ; and the student is carried lorward, by eacy and 
gradual steps, over the whole field of mathematical inquiry, and that, too, in a 
shorter time than is usually occupied in mastering a single department. I sincere- 
ly and heartily recommend, them to the attention of my fellow-teachers in Canada. 

From D. W. Steele, Prin. Philekoian Academy, Cold Springs, Texas. 
I have used Davies' Arithmetics till I know them nearly by heart. A better 
series of school-books never were published. I have recommended them until 
they are now used in all this region of country. 



A large mass of similar " Opinions " may be obtained by addressing the pub 
Ushers for special circular for Davies' Mathematics. New recommendations ar* 
pablished in current numbers of the Educational Bulletin. 

19 



JVatio7ial Series of Standard Sc?iool-!Books* 

DAVIES' NATIONAL COURSE of MATHEMATICS, 

ITS RECORD. 

In claiming for this series the first place among American text-books, of what 
ever class, the Publishers appeal to the magnificent record which its volumes 
have earned during the thirty-five years of Dr. Charles Davies' mathematical 
labors. The unremitting exertions of a life-time have placed the modern series oil 
the same proud eminence among competitors that each of its predecessors h&i 
successively enjoyed in a course of constantly improved editions, now rounded tc 
their perfect fruition — for it seems almost that this science is susceptible of n£ 
farther demonstration. 

During the period alluded to, many authors and editors in this department have 
started into public notice, and by borrowing ideas and processes original with Dr. 
Davies, have enjoyed a brief popularity, but are now almost unknown. Many of 
the series of to-day, built upon a similar basis, and described as " modern books," 
are destined to a similar fate ; while the most far-seeing eye will find it difficult to 
fix the time, on the basis of any data afforded by their past history, when these 
books will cease to increase and prosper, and fix a still firmer held on the affectiun 
of every educated American. 

One cause of this unparalleled popularity is found in the fact that the enterprise 
of the author did not cease with the original completion of his books. Always a 
practical teacher, he has incorporated in his text-books from time to time the ad- 
vantages of every improvement in methods of teaching, and every advance in 
science. During all the years in which he has been laboring, he constantly sub- 
mitted his own theories and those of others to the practical test of the class-room 
—approving, rejecting, or modifying them as the experience thus obtained might 
suggest. In this way he has been able to produce an almost perfect series of 
class-books, in which every department of mathematics has received minute and 
exhaustive attention. 

Nor ha& he yet retired from the field. Still in the prime of life, and enjoying a 
ripe experience which no other living mathematician or teacher can emulate, his 
pen is ever ready to carry on the good work, as the progress of science may de- 
mand. Witness his recent exposition of the "Metric System," which received 
the official endorsement of Congress, by its Committee on Uniform Weights and 
Measures. 

Davies' System is the acknowledged National Standard for the 
United States, for the following reasons :— 

1st. It is the basis of instruction in the great national schools at West Point 
and Annapolis. 

2d. It has received the quasi endorsement of the National Congress. 

3d. It is exclusively used in the public schools of the National Capital. 

4th. The officials of the Government use it as authority in all cases involving 
mathematical questions. 

5th. Our great soldiers and sailors commanding the national armies and navies 
were educated in this system. So have been a majority of eminent scientists in 
this country. All these refer to "Davies " as authority. 

6th. A larger number of American citizens have received their education from 
this than from any other series. 

7th. The series has a larger circulation throughout the whole country than any 
©ther, being extensively used in every State in the Union,. 

20 



The National Series of Standard SchoolSooks. 

MATHEMATICS-Continued. 

PECK'S ARITHMETICS. 

By the Prof, of Mathematics at Columbia College, New York. 

i. Peck's First Lessons in Numbers, • ■ • $o 25 

Embracing all that is usually included in what are called Primary and Intel- 
lectual Arithmetics; proceeding gradually from object lessons to abstract num- 
bers; developing Addition and Subtraction simultaneously: with other attrac- 
tive novelties. 

2. Peck's Manual of Practical Arithmetic, • so 

An excellent " Brief" course, conveying a sufficient knowledge of Arithmetic 
for ordinary business purposes. 

It is thoroughly " practical," because the author believes the Theory cannot be 
studied with advantage until the pupil has acquired a certain facility in combin- 
ing numbers, which can only be had by practice. 

3. Peck's Complete Arithmetic, 90 

The whole subject— theory and practice — presented within very moderate 
limits. This author's most remarkable faculty of mathematical treatment is 
comprebended in three words : System, Conciseness, Lucidity. The directness 
and simplicity of this work cannot be better expressed than in the words of a 
correspondent who adopted the book at once, because, as he said, it is " free 
from that juggling with numbers' 1 '' practiced by many authors. 

From the " Galaxy" Neiv York. 

In the " Complete Arithmetic" each part of the subject is logically developed. 
First are given the necessary definitions ; second, the explanations of such signs 
(if any) as are used ; third, the principles on which the operation depends ; 
fourth, an exemplification of the manner in which the operation is performed, 
which is so conducted that the reason of the rule which is immediately thereafter 
deduced is made perfectly plain ; after which follow numerous graded examples 
and corresponding practical problems. All the parts taken together are arranged 
in logical order. The subject is treated as a whole, and not as if made up of 
segregated parts. It may seem a simple remark to make that (for example) addi- 
tion is in principle one and the same everywhere, whether employed upon simple 
or compound numbers, fractions, etc., the only difference being in the unit in- 
volved ; but the number of persons who understand this practically, compared to 
the number who have studied arithmetic, is not very great. The student of the 
" Complete Arithmetic " cannot fail to understand it. All the principles of the 
science are presented within moderate limits. Superfluity of matter— to supple- 
ment defective definitions, to make clear faulty demonstrations and rules ex- 
pressed either inaccurately or obscurely, to make provision for a multiplicity of 
cases for which no provision is requisite — has been carefully avoided. The 
definitions are plain and concise ; the principles are stated clearly and accurately ; 
the demonstrations are full and complete ; the rules are perspicuous and compre- 
hensive ; the illustrative examples are abundant and well fitted to familiarize the 
student with the application of principles to the problems of science and of 
every-day life. 

J^g^ The Definitions constitute the power of the book. We have never seen 
them excelled for clearness and exactness.— Iowa School Journal. 

21 



7~he National Series of Standard School-Sootes. 

PENMANSHIP. 

* — » 

Beers' System of Progressive Penmanship. 

Per dozen $1 68 

This " round hand " system of Penmanship in twelve numbers, com- 
mends itself by its simplicity and thoroughness. The first four numbers 
are primary books. Nos. 5 to 7, advanced books for boys. Nos. 8 to 10, 
advanced books for girls. Nos. 11 and 12, ornamental penmanship. 
These books are printed from steel plates (engraved by McLees), and are 
unexcelled in mechanical execution. Large quantities are annually sold. 

Beers' Slated Copy Slips, per set *50 

All beginners should practice, for a few weeks, slate exercises, familiar- 
izing them with the form of the letters, the motions of the hand and arm, 
&c, &c. These copy slips, 32 in number, supply all the copies found in a 
complete seiies of writing-books, at a trifling cost. 

Payson,Dunton&Scribner'sCopy-B'ks.P-doz*i 80 

The National System of Penmanship, in three distinct series — (11 Com- 
mon School Series, comprising the first six numbers ; (2) Business Series, 
Nos. 8, 11, and 12 ; (3) Ladies 1 Series, Nos. 7, 9, and 10. 

Fulton & Eastman's Chirographic Charts,*3 75 

To embellish the school room walls, and furnish class exercise in the 
elements of Penmanship. 

Payson's Copy-Book Cover, per hundred .*4 oo 

Protects every page except the one in use, and furnishes "lines" with proper 
slope for the penman, under. Patented. 

National Steel Pens, Card with all kinds . . . *15 

Pronounced by competent judges the perfection of American-made pens, and 
superior to any foreign article. 



SCHOOL SERIES. 
School Pen, per gross, . .$60 
Academic Pen, do . . 63 

Pine Pointed Pen, per gross 70 

POPULAR SERIE3. 

Capitol Pen, per gross, . . 1 00 

do do pr. box of 2 doz. 25 

Bullion Pen (imit. gold) pr. gr. 75 

Ladies' Pen do 63 



Index Pen, per gross ... 75 
BUSINESS SERIES. 



Albata Pen, per gross, 
Bank Pen, do 
Empire Pen, do 
Commercial fen, per gross 
Express Pen, do 

Falcon Pen, do 

Elastic Pen, do 



40 
TO 
70 
60 
75 
70 
T5 



Stimpson's* Scientific Steel Pen, per gross .*2 oo 

One forward and two backward arches, ensuring great strength, well- 
balanced elasticity, evenness of point, and smoothness of execution. One 
gross in twelve contains a Scientific Gold Pen. 

Stimpson's Ink-Retaining Holder, per doz. .*2 oo 

A simple apparatus, whic-. does not get out of order, withholds at a 
single dip as much ink as the peD would otherwise realize from a dozeu 
trips to the inkstand, which it supplies with moderate and easy flow. 

Stimpson's Gold Pen, $3 oo ; with Ink Retainer* 4 50 
Stimpson's Penman's Card, * so 

One dozen Steel Pens (assorted points) and Patent Ink-retaining Pen 
holder. 

22 



The JVallonat Series of Standard Schooi-jBooks. 



HISTORY. 



Monteith's Youth's History, ..$75 

A History of the United States for beginners. It is arranged upon the 
catechetical plan, with illustrative maps and engravings, review questions, ';' 

dates in parentheses (that their study may be optional with the younger 
elass of learners), and interesting biographical Sketches of all persons 
who have been prominently identified with the history of our country, 

Wiiiard's United States, School edition, ... i 40 

Do. do. University edition, . 2 25 

The plan of this standard work is chronologically exhibited in front of 
the title-page ; the Maps and Sketches are found useful assistants to the 
memory, and dates, usually so difficult to remember, are so systematically 
arranged as in a great degree to obviate the difficulty. Candor, impar- 
tiality, and accuracy, are the distinguishing features of the narrative 
portion. 

Willard's Universal History, 2 25 

The most valuable features of the " United States" are reproduced in 
this. The peculiarities of the work are its great conciseness and the 
prominence given to the chronological order of events. The margin 
marks each successive era with great distinctness, so that the pupil re- 
tains not only the event but its time, and thus fixes the order of history 
firmly and usefully in his mind. Mrs. Willard's books are constantly 
revised, and at all times written up to embrace important historical 
events of recent date. 

Berard's History of England, i 

By an authoress well known for the success of her History of the United 
States. The social life of the English people is felicitously interwoven, 
as in fact, with the civil and military transactions of the realm. 

Ricord's History of Rome, i 7 ^ 

Possesses the charm of an attractive romance. The Fables with which 
this history abounds are introduced in such a way as not to deceive the 
inexperienced, while adding materially to the value of the work as a reli- 
able index to the character and institutions, as well as the history of the 
Koman people. 

Hanna's Bible History, i 25 

The only compendium of Bible narrative which affords a connected and 
chronological view of the important events there recorded, divested of all 
superfluous detail. 

Nummary of History, Complete 60 

American History, $0 40. French and Eng. Hist. 35 

A well proportioned outline of leading events, condensing the substance of the 
more extensive text-book in common use into a series of statements so brief, th&t 
every word may be committed to memory, and yet so comprehensive that it 
presents an accurate though general view of the whole continuous life of natioua. 

Marsh's Ecclesiastical History, 2 00 

Questions to ditto, 75 

Affording the History of the Church in all ages, with accounts of the 
pagan world during Biblical periods, and the character, rise, and progress 
of all Religions, as well as the various sects of the worshipers of Christ. 
The work is entirely non-sectarian, though strictly catholic. 

Mill's History of the Jews, 175 

23 



The National Series of Standard School- jBooks. 

HISTORY-Continued. 

BARNES' ONE-TERM HISTORY. 
A Brief History of the United States, ■ . .$1 50 

This is probably the most original school-book published, for many years, 
in any department. A few of its claims are the following : 

1. Brevity- — The text is complete for Grammar School or intermediate 
classes, in 290 12mo pages, large type. It may readily be completed, if desired, in 
one term of study. 

2. Comprehensiveness Though bo brief, this book contains the pith of a*: 

the wearying contents of the larger manuals, and a great deal more than the mem- 
ory usually retains from the latter. 

3. Interest has been a prime consideration. Small books have heretofore 
been bare, full of dry statistics, unattractive. This one is charmingly written, 
replete with anecdote, and brilliant with illustration. 

4. Proportion Of Events.— It is remarkable for the discrimination with 
which the different portions of our history are presented according to their im- 
portance. Thus the older works being already large books when the civil war 
took place, give it less space than that accorded to the Revolution. 

5. Arrangement. — In six epochs, entitled respectively, Discovery and Settle- 
ment, the Colonies, the Revolution, Growth of States, the Civil War, and Current 
Events. 

6. Catch Words. — Each paragraph is preceded by its leading thought in 
prominent type, sending in the student's mind for the whole paragraph. 

7. Key Not3S,— Analogous with this is the idea of grouping battles, etc., 
about some central event, which relieves the sameness so common in such de- 
scriptions, and renders each distinct by some striking peculiarity of its own. 

8. Foot Notes. — These are crowded with interesting matter that is not 
strictly a part of history proper. They may be learned or not, at pleasure. They 
are certain in any event to be read. 

9. Biographies of all the leading characters are given in full in foot-notes. 

10. Maps. — Elegant and distinct Maps from engravings on copper-plate, and 
beautifully colored, precede each epoch, and contain all the places named. 

11. Questions are at the back of the book, to compel a more independent use 
f f the text. Both text and questions are so worded that the pupil must give in- 
t illigent answers in his own wobds. " Yes " and "No " will not do. 

12. Historical Recreations.— These are additional questions to test the stu- 
dent's knowledge, in review, as : " What trees are celebrated in our history ?" 
" When did a fog save our army ? " " What Presidents died in office ? " " When 
was the Mississippi our western boundary ? " " Who said, ' I would rather be 
right than President ? ' " etc. 

13. The Illustrations, about seventy in number, are the work of our best 
artists and engravers, produced at great expense. They are vivid and interest- 
ing, and mostly upon subjects never before illustrated in a school-book. 

14. Dates.— Only the leading dates are given in the text, and these are so 
associated as to assist the memory, but at the head of each page is the date of the 
event first mentioned, and at the rlose of each epoch a summary of events and dates. 

15. The Philosophy of History is studiously exhibited— the causes and 
effects of events being distinctly traced and their interconnection shown. 

16. Impartiality. — All sectional, partisan, or denominational views are 
avoided. Facts are stated after a careful comparison of all authorities without 
the least prejudice or favor. 

17. Index.— A verbal index at the close of the book perfects it as a work of 
reference. 

It will be observed that the above are all particulars in which School Histories 
have been signally defective, or altogether wanting. Many other claims to iavor 
U shares in common with its predecessors. 

24 






The National Series of Standard Sc?iool-'Books. 

HISTORY-Contmued. 

Hunter's Historical Games, witli cards . . . 10 75 

An invaluable accompaniment for the text-book, by way of stimulating interest 
in the Class ; affording, at once, Amusement and Instruction. 

SOME TESTIMONIALS FOR BARNES' BRIEF HISTORY. 

From Hon. J. M. McKenzie, Supt. Pub. Inst., Nebraska. 
I have examined your "Brief History of the United States," and like it real well ; 
and were I teaching a graded school, I think I should use it as a text-book. 

From Hon. H. B. Wilson, Supt. Pub. Inst., Minnesota. 
I have read with much interest the " One-Term History of the United States." I 
am much pleased with it. In my judgment, it contains all of the United States his- 
tory that the majority of pupils in our common schools can spare time to study. 

From Pres. Edward Brooks, Millersville State Normal School, Pa. 
It is a work that will be a favorite with teachers and pupils. Its scope and style 
especially adapt it for use in our public schools. I cordially commend it to teachers 
desiring to introduce an interesting and practical text-book upon this subject. 

From Prep. Barker, Buffalo State Normal School. N Y. 
In the copy of your "Brief History," before me, the important items to be learned 
in history seem most ingeniously brought out and kept in the foreground. These 
items are time, persons, places, and events. It has the appearance of an exceedingly 
fresh and systematic work. I think I shall put it into my classes. 

From Prop. Wm. F. Allen, State Univ. of Wisconsin. 
I think the author of the new " Brief History of the United States " has been very 
successful in combining brevity with sufficient fullness and interest. Particularly, 
he has avoided the excessive number of names and dates that most histories con- 
tain. Two features that I like very much are the anecdotes at tha foot, of the page 
and the " Historical Beereaiions " in the Appendix. The latter, I think, is quite a 
new feature, and the other is very well executed. 

From S. G. Wright, Assist.-Supt. Pub. Inst., Kansas. 
It is with extreme pleasure we submit our recommendation of the "Brief History 
of the United States." It meets the needs of young and older children, combining 
concision with perspicuity, and if "brevity is the soul of wit," this "Brief His- 
tory" contains not only that well-chosen ingredient, but Avisdom sufficient to en- 
lighten those students who are wearily longing for a " new departure " from certain 
old and uninteresting presentations of fossilized writers. We congratulate a pro- 
gressive public upon a progressive book. 

From Hon. Newton Bateman, Supt. Pub. Inst., PMnois. 
Barnes 1 One-Term History of the United States is an exceedingly attractive and 
spirited little book. Its claim to several new and valuable features seems well 
founded. Under the form of six well-defined Epochs, the History of the United 
States is traced tersely, yet pithily, from the earliest times to the present day. A 
good map precedes each epoch, whereby the history and geography of the period 
may be studied together, as they ahvays should be. The syllabus of each paragraph 
is made to stand in such bold relief, by the use of large, heavy type, as to be of 
much mnemonic value to the student. The book is written in a sprightly and 
piquant style, the interest never flagging from beginning to end — a rare and difficult 
achievement in works of this kind. 

From the " Chicago Schoolmaster'''' {Editorial). 
A thorough examination of Barnes 1 Brief History of the United States brings the 
examiner to the conclusion that it is a superior book in almost every respect. The 
book is neat in form, and of good material. The type is clear, large, and distinct. 
Th3 facts and dates are correct. The arrangement of topics is just the thing needed 
in a history text-book. By this arrangement the pupil can see at once what he is 
expected to do. The topics are well selected, embracing the leading ideas or prin- 
cipal events of American history. . . . The book as a whole is much superior 
to any I have examined. So much do I think this, that I have ordered it for my 
class, and shall use it in my school. (Signed) B. W. Baker. 

Baker's Brief History of Texas, $1 25 

25 



The National Series of Standard School-Books. 

DRAWING. 



Chapman's American Drawing Book, . . .*I6 oo 

The standard American text-book and authority in all branches of art. A com* 

?)ilation of art principles. A manual for the amateur, and basis of study for the pro* 
essional artist. Adapted for schools and private instruction. 

Contents.— "Any one who can Learn to Write can Learn to Draw." — Primary 
Instruction in Drawing.— Rudiments of Drawing the Human Head. — Rudiments in 
Drawing the Human Figure. — Rudiments of Drawing.— The Elements of Geometry.— 
Perspective.— Of Studying and Sketching from Nature. — Of Painting. — Etching and 
Engraving.— Of Modeling.— Of Composition — Advice to the American Art-Student. 
The work is of course magnificently illustrated with all the original designs. 

Chapman's Elementary Drawing Book, . . l 50 

A Progressive Course of Practical Exercises, or a text-book for the training of the 
eye and hand. It contains the elements from the larger work, and a copy should 
be in the hands of every pupil; while a copy of the "American Drawing Book," 
named above, should be at hand for reference by the class. 

The Little Artist's Portfolio, *50 

25 Drawing Cards (progressive patterns), 25 Blanks, and a fine Artist's Pencil, 
all in one neat envelope. 

Clark's Elements of Drawing, *i oo 

A complete course in this graceful art, from the first rudiments of outline to the 
finished sketches of landscape and scenery. 

Fowle's Linear and Perspective Drawing, • *60 

For the cultivation of the eye and hand, with copious illustrations and direc- 
tions for the guidance of the unskilled teacher. 

Rlonk's Drawing Books— Six Numbers, per set, *2 25 

Eacu "cook contains eleven large patterns, with opposing blanks. No. 1. Elemen- 
tary Studies. No. 2. Studies of Foliage. No. 3. Landscapes. No. 4. Animals, L 
No. 5. Animals, II. No. 6. Marine Views, etc. 

Allen's Map-Drawing, . . . 25cts.; Scale, 25 

This method introduces a new era in Map-Drawing, for the following reasons :— 
1. It is a system. This is its greatest merit.— 2. It is easily understood and taught. 
—3. The eye is trained to exact measurement by the use of a scale.— 4. By no spe- 
cial effort of the memory, distance and comparative size are fixed in the mind.— 
5. It discards useless construction of lines.— 6. It can be taught by any teacher, even 
though there may have been no previous practice in Map-Drawing.— 7. Any pupil 
old enough to study Geography can learn by this System, in a short time, to draw 
accurate maps. — 8. The System is not the result of theory, but comes directly from 
the school-room. It has been thoroughly and successfully tested there, with all 
grades of pupils. — 9. It is economical, as it requires no mapping plates. It gives 
the pupil the ability of rapidly drawing accurate maps. 

Ripley's Map-Drawing, 1 25 

Based on the Circle. One of the most efficient aids to the acquirement of a 
knowledge of Geography is the practice of map-drawing. It is useful for the same 
reason that the best exercise in orthography is the writing of difficult words. 
Sight comes to the aid of hearing, and a double impression is produced upon the 
memory. Knowledge becomes less mechanical and more intuitive. The student 
who has sketched the outlines of a country, and dotted the important places, is little 
likely to forget either. The impression produced may be compared to that of a 
traveller who has been over the ground, while more comprehensive and accurate in 
detail. 

26 



The National Series of Standard School-Books. 

BOOK-KEEPING. 



Folsom's Logical Book-keeping, * ... .$200 
Folsom's Blanks to Book-keeping, ...... *4 50 

This treatise embraces the interesting and important discoveries 
of Prof. Folsom (of the Albaiiy " Bryant & Stratton College"), the par- 
tial enunciation of which in lectures and otherwise has attracted so 
much attention in circles interested in commercial education. 

After studying business phenomena for many years, he has arrived 
at the positive laws and principles that underlie the whole subject of 
Accounts ; finds that the science is based in Value as a generic term • 
that value divides into two classes with varied species ; that all the 
exchanges of values are reducible to nine equations ; and that all the 
results of all these exchanges are limited to thirteen in number. 

As accounts have been universally taught hitherto, without setting 
out from a radical analysis or definition of values, the science has 
been kept in great obscurity, and been made as difficult to impart as 
to acquire. On the new theory, however, these obstacles are chiefly 
removed. In reading over the first part of it, in which the governing 
laws and principles are discussed, a person with ordinary intelligence 
will obtain a fair conception of the double entry process of accounts. 
But when he comes to study thoroughly these laws and principles as 
there enunciated, and works out the examples and memoranda which 
elucidate the thirteen results of business, the student will neither fail 
in readily acquiring the science as it is, nor in becoming able intelli- 
gently to apply it in the interpretation of business. 

Smith & Martin's Book-keeping, l 25 

Smith & Martin's Blanks, *60 

This work is by a practical teacher and a practical book-keeper. 
It is of a thoroughly popular class, and will be welcomed by every 
one who loves to see theory and practice combined in an easy, con- 
cise, and methodical form. 

The Single Entry portion is well adapted to supply a want felt in 
nearly all other treatises, which seem to be prepared mainly for the 
use of wholesale merchants, leaving retailers, mechanics, farmers, 
etc., who transact the greater portion of the business of the country, 
without a guide. The work is also commended, on this account, for 
general use in Young Ladies' Seminaries, where a thorough ground- 
ing in the simpler form of accounts will be invaluable to the future 
housekeepers of the nation. 

The treatise on Double Entry Book-keeping combines all the ad- 
vantages of the most recent methods, with the utmost simplicity of 
application, thus affording the pupil all the advantages of actual ex- 
perience in the counting-house, and giving a clear comprehension of 
the entire subject through a judicious course of mercantile trans- 
actions. 

The shape of the book is such that the transactions can be pre- 
sented as in actual practice ; and the simplified form of Blanks — 
three in number— adds greatly to the ease experienced in acquiring 
the science. 

27 



The National Series of Standard Sc7iool-!Boofos. 

NATURAL SCIENCE. 



FAMILIAR SCIENCE. 
Norton & Porter's First Book of Science, • ti 75 

By eminent Professors of Yale College. Contains the principles of Natural 
Philosophy, Astronomy, Chemistry, Physiology, and Geology. Arranged on the 
Catechetical plan for primary classes and beginners. 

Chambers' Treasury of Knowledge, .... 1 ^5 

Progressive lessons upon— -first, common things which lie most , immediately 
iround us, and first attract the attention of the young mind ; second, common objects 
from the Mineral, Animal, and Vegetable kingdoms, manufactured articles, and 
miscellaneous substances ; third, a systematic view of Nature under the various 
sciences. May be used as a Reader or Text-book. 

NATURAL PHILOSOPHY. 
Norton's First Book in Natural Philosophy, l oo 

By Prof. Norton, of Yale College. Designed for beginners. Profusely illustrated . 
and arranged on the Catecheticarplan. 

Peck's Ganot's Course of Nat. Philosophy, . l 75 

The standard text-book of France, Americanized and popularized by Prof. Peck, 
of Columbia College. The most magnificent system of illustration ever adopted in 
an American school-book is here found. For intermediate classes. 

Peck's Elements of Mechanics, 2 00 

A suitable introduction to Bartlett's higher treatises on Mechanical Philosophy, 
and adequate in itself for a complete academical course. 

Bartlett's synthetic, ad analytic, Mechanics, • each 5 00 
Bartlett's Acoustics and Optics, 3 50 

A system of Collegiate Philosophy, by Prof. Bartlett, of West Poiut Military 
Academy. 

Steele's 14 Weeks Course in Philos. (see P . 34) 1 50 
Steele's Philosophical Apparatus, • • • • *125 00 

Adequate to performing the experiments in the ordinary text-books. The articles 
will be sold separately, if desired. See special circular for details. 

GEOLOGY. 
Page's Elements of Geology, 1 25 

A volume of Chambers' Educational Course. Practical, simple, and eminently 
calculated to make the study interesting. 

Emmons' Manual of Geology, 1 25 

The first Geologist of the country has here produced a work worthy of his repu- 
tation. 

Steele's 14 Weeks Course (seep. 34) 1 so 

Steele's Geological Cabinet, *40 og 

Containing 125 carefully selected specimens. In four parts. Sold separately, U 
desired. See circular for details. 

28 



The JVatio?ia'o Series of Sta?iaard School-liook£* 



Peck's Ganot's Popular Physics. 

TESTIMONIALS. 

From Prof. Aloxzo Colun, Cornell College, Iowa. 
I am pleased with. it. I have decided to introduce it as a text-book. 

From H. F. Johnson, President Madison College, Sharon, Miss.. 
I am pleased with Peck's Ganot, and think it a magnificent hook. 

From Prop. Edward Brooes, Pennsylvania State Formal Softool. 

So eminent are its merits, that it will he introduced as the text-hook upon el3 
mentary physics in this institution. 

From H. H. Locewood, Professor Natural Philosophy U. S. Naval Academy. 

I am so pleased with it that I will probably add it to a course of lecturer given tQ 
the midshipmen of this school on physics. 

From Geo. S. Macete, Professor Natural EHiory University of Nashville, Tenn. 

I have decided on the introduction of Pecks Ganot's Pnilosophy, as I am satis- 
fied that it is the best book for the purposes of my pupils that I have seen, con** 
Dining simplicity of explanation with elegance of illustration. 

From W. S. McEae, Superintendent Vevay Public Schools, Indiana. 

Having carefully examined a number of text-books on natural philosophy, I da 
not hesitate to express my decided opinion in favor of Peck's Ganot. The matter, 
style, and illustration eminently adapt the work to the popular wants. 

FromHsY. Samuel McKixney, D.D., PresH Austin College, Huntsville, Texas. 

It gives me pleasure to commend it to teachers. I have taught some classes with 
it as our text, and must say, for simplicity of style and clearness of illustration, J 
have found nothing as yet published of equal value to the teacher and pupil. 

From C. V. Spear, Principal Maplewood Institute, Pittsfield, Mass. 

I am much pleased Avith its ample illustrations by plates, and its clearness and 
simplicity of statement. It covers the ground usually gone over by our higher 
classes, and contains many fresh illustrations from life or daily occurrences,"and 
new applications of scientific principles to such. 

From J. A. Bantteld, Superintendent Marshall Public Schools, Michigan. 

I have used Peck's Ganot since 1863, and with increasing pleasure and satisfac- 
tion each term. I consider it superior to any other work on physics in its adapta- 
tion to our high schools and. academies. Its illustrations are superb — better 
than three times their number of pages of fine print. 

From A. Scheyler, Prof, of Mathematics in Baldwin University, Berea, Ohio. 

After a careful examination of Peck's Ganot's Natural Philosophy, and an actual 
rest of its merits as a text-book, I can heartily recommend it as admirably adapted 
to' meet the wants of the grade of students for which it is intended. Its diagrams 
and illustrations are unrivaled. We use it in the Baldwin University. 

From D. C. Yak Nobsian-, Principal Van Norman Institute, New York. 
The Natural Philosophy of M. Ganot. edited by Prof. Peck, is, in my opinion, 
fhe best work of its kind, for the use intended, ever published in this country. 
Whether regarded in relation to the natural order of the topics, the precision and 
clearness of its definitions, or the fullness and beauty of its illustrations, ii is cer< 
tainly, I think, an advance. 



For many similar testimonials, see current numbers of the HinBtrated Ed 
ideational BoJietin. 

29 



The JVatio7ial Series of Standard School- !Sooks. 

NATURAL SCIENCE-Continued. 

CHEMISTRY. 

Porter's First Book of Chemistry, . . . . $l oo 
Porter's Principles of Chemistry, 2 oo 

The above are widely known as the productions of one of the most eminent scien- 
tific men of America. The extreme simplicity in the method of presenting the 
science, while exhaustively treated, has excited universal commendation. 

Darby's Text-Book of Chemistry, 1 75 

Purely a Chemistry, divesting the subject of matters comparatively foreign to it 
(such as heat, light, electricity, etc.), but usually allowed to engross too much attem 
tion in ordinary school-books. 

Gregory's Organic Chemistry, 2 50 

Gregory's Inorganic Chemistry, 2 50 

The science exhaustively treated. For colleges and medical students. 

Steele's Fourteen Weeks Course, 1 so 

A successful effort to reduce the study to the limits of a single term, thereby 
making feasible its general introduction in institutions of every character. The 
author's felicity of style and success in making the science pre-eminently interest- 
ing are peculiarly noticeable features. (See page 34.) 

Steele's Chemical Apparatus, *20 oo 

Adequate to the performance of all the important experiments. 

BOTANY. 
Thinker's First Lessons in Botany, .... 40 

For children. The technical terms are largely dispensed with in favor of an 
easy and familiar style adapted to the smallest learner. 

Wood's Object-Lessons in Botany, . . . . l 50 
Wood's American Botanist and Florist, . . 2 50 
Wood's New Class-Book of Botany, .... 3 50 

The standard text-books of the United States in this department. In style they 
are simple, popular, and lively ; in arrangement, easy and natural ; in description, 
graphic and strictly exact. The Tables for Analysis are reduced to a perfect sys- 
tem. More are annually sold than of all others combined. 

Wood's Plant Record, *75 

A simple form of Blanks for recording observations in the field. 

Wood's Botanical Apparatus, *8 oo 

A portable Trunk, containing Drying Press, Knife, Trowel, Microscope, and 
Tweezers, and a copy of Wood's Plant Record — composing a complete outfit for 
the collector. 

Young's Familiar Lessons, 2 00 

Darby's Southern Botany, 2 oo 

Embracing general Structural and Physiological Botany, with vegetable products, 
and descriptions of Southern plants, and a complete Flora of the Southern States, 

30 



The National Series of Standard School-Books. 



WOOD'S BOTANIES. 



From Pres. R. B. Burleson, Waco University, Texas. 
Wood's Botanies— books that meet every want in their line. 

From Prist. J. G. Ralston, Norristown Seminary, Pa. 
We find the •' Class-Book " entirely satislactory. 

From Pres. D. F. Bittle, JRoanoke College, Va. 
Your text-books on Botany are the best for students. 

From Prof. W. C. Pierce, Baldwin University, Ohio. 
I thUk his Flora the best we have. His method of analysis is excellent. 

From Prof. Blakeslee, State Normal School, Potsdam, N. Y. 
Tt is admirably concise, yet it does not seem to be deficient or obscure. In paper, 
print, and binding, the book leaves little to be desired. 

From Pres. J. M. Gregory, State Agricultural College, HI. 
I find myself greatly pleased with the perspicuity, compactness, and complete- 
ness of the book (Wood's Botanist and Florist). I shall recommend it freely to my 
friends. 

From Prof. A. Wlnchell, University of Michigan. 
I am free to say that I had been deeply impressed, I may say almost astonished, 
at the evidences which the work bears of skillful and experienced authorship in 
this field, and nice and constant adaptation to the wants and conveniences of 
students of Botany. I pronounce it emphatically an admirable text-book. 

From Prof. Richard Owen, University of Indiana. 
I am well pleased with the evidence of philosophical method exhibited in the 
general arrangement, as well as with the clearness of the explanations, the ready 
intelligibility of the analytical tables, and the illustrative aid furnished by the 
numerous and excellent wood-cuts. I design using the work as a text-book with 
my next class. 

From Prin. B. R. Anderson, Columbus Union School, Wisconsin. 
I have examined several works with a view to recommending some good text- 
book on Botany, but I lay them all aside for " Wood's Botanist and Florist." The 
arrangement of the book is in my opinion excellent, its style fascinating and attrac- 
tive, its treatment of the various departments of the science is thorough, and last, 
but far from unimportant, I like the topical form of the questions to eacli chapter. 
It seems to embrace the entire science. In fact, I consider it a complete, attractive, 
and exhaustive work. 

From M. A. Marshall, New Haven High School, Conn. 
It has all the excellencies of the well-known Class-Book of Botany by the same 
author in a smaller book. By a judicious system of condensation, the size of the 
Flora is reduced one-half, while no species are omitted, and many new ones are 
added. The descriptions of species are very brief, yet sufficient to identify the 
plant, and, when taken in connection with the generic description, form a complete 
description of the plant. The book as a whole will suit the wants of classes better 
than anything I have yet seen. The adoption of the Botanist and Florist would 
not require the exclusion of the Class-Book of Botany, as they are so arranged that 
both might be used by the same class. 

From Prof. G. H. Perkins, University of Vermont and State Agricultural College. 
I can truly say that the more I examine Wood's Class-Book, the better pleased I 
am with it. In its illustrations, especially of particulars not easily observed by the 
student, and the clearness and compactness of its statements, as well as in the ter- 
ritory its flora embraces, it appears to me to surpass any other work I know of. 
The whole science, so far as it can be taught in a college course, is well presented, 
and rendered unusually easy of comprehension. The mode of analysis is excellent, 
avoiding as it does to a great extent those microscopic characters which puzzle the 
beginner, and using those that are obvious as far as possible. I regard the work as 
a most admirable one, and shall adopt it as a text-book another year. 

31 



The JVatlonal Series of Standa?'d School-Hooks* 

NATURAL SCIENCE-Continued. 

PHYSIOLOGY. 
Jarvis' Elements of Physiology, • . " • .. .$ 75 
Jarvis' Physiology and Laws of Health, . 1 65 

The only books extant which approach this subject with a proper view 
of the tru« object of teaching Physiology in schools, viz., that scholars ( 

may know how to take care of their own health. In bold eontrast with 
the abstract Anatomies, which children learn as they would Greek or 
Latin (and forget as soon), to discipline the mind, are these text-books, 
using the science as a secondary consideration, and only so far as is 
necessary for the comprehension of the laivs of health. 

Hamilton's Vegetable & Animal Physiology, 1 25 

The two branches of the science combined in one volume lead the stu- 
dent to a proper comprehension of the Analogies of Nature. 

Steele's Fourteen Weeks Course (see p. 34), . l so 

ASTRONOMY. 
Steele's Fourteen Weeks' Course, i 50 

Reduced to a single term, and better adapted to school use than any 
work heretofore published. Not written for the information of scientific 
men,_ but for the inspiration of youth, the pages are not burdened with a 
multitude of figures which no memory could possibly retain. The whole 
subject is presented in a clear and concise form. (See p. 34.) 

Willard's School Astronomy, 1 oo 

By means of clear and attractive illustrations, addressing the eye in 
many cases by analogies, careful definitions of all necessary technical 
terms, a caieful avoidance of verbiage and unimportant matter, particular 
attention to analysis, and a general adoption of the simplest methods, 
Mrs. Willard has made the best and most attractive elementary Astron- 
omy extant. 

Mclntyre's Astronomy and the Globes, • - 1 50 

A complete treatise for intermediate classes. Highly approved. 

Bartlett's Spherical Astronomy, 5 oo 

The West Point course, for advanced classes, with applications to the 
current wants of Navigation, Geography, and Chronology. 

? NATURAL HISTORY. 

Carll's Child's Book of Natural History, • • o 50 

Illustrating the Animal, Vegetable, and Mineral Kingdoms, with appli- 
cation to the Arts. For beginners. Beautifully and copiously illustrated. 

ZOOLOGY. 
Chambers' Elements of Zoology, 1 50 

A complete and comprehensive system of Zoology, adapted for aca- 
demic instruction, presenting a systematic view of the Animal Kingdom 
as a portion of external Nature. 

32 



National Series of Standard School- Books. 



Jams' Physiology and Laws of Health. 

TESTIMONIALS. 

From Samuel B. McLane, Superintendent Public Schools, Keokuk s Iowa. 
I am glad to see a really good text-book on this much neglected branch. This ia 
•jlear, concise, aceurate, and eminently adapted to the clans-room. 

From William F. Wyebs, Principal of Academy, West Chester, Pennsylvania. 

A thorough examination has satisfied me of its superior claims as a text-book to tbe 
$ttention of teacher and taught. I shall introduce it at once. 

From H. R. Sanfobo, Principal of East Genesee Conference Seminary, N. Y. 

"Jarvis' Physiology" is received, and fully met our expectations. We immediately 
adopted it. 

From Isaac T. Goodnow, State Superintendent of Kansas— published in conrieetion 
with the " School haw" 

" Jarvis' Physiology," a common-sense, practical work, with just enough of anat- 
omy to understand the physiological portions. The last six pages, on Man's Respon 
sibUity for his own health, are worth the price of the book. 

From D. W. Stevens, Superintendent Public Schools, Fall River f Mass. 

I have examined Jarvis' " Physiology and Laws of Health," which you had the 
kindness to send to me a short time ago. In my judgment it is far the best work of 
the kind within my knowledge. It has been adopted as a text-book in our public 
schools. 

From Heney G. Denny, Chairman Book Committee, Boston, Mass. 

The very excellent " Physiology " of B_. Jarvis I had introduced into our High 
School, where the 6tudy had been temporarily dropped, believing it to be by far the 
best work of the kind that had come under my observation; indeed, the reintroduc- 
tion of the study was delayed for some months, because Dr. Jarvis' book could not be 
had, and we were unwilling to take any other. 

From Peof. A. P. Peabodt, D.D., LL.D., Harvard University. 
* * I have been in the habit of examining school-books with great care, and I 
hesitate not to say that, of all the text-books on Physiology which have been given to 
the public, Dr. Jarvis' deserves the first place on the score of accuracy, thoroughness, 
method, simplicity of statement, and constant reference to topics of practical interest 
and utility. 

From James N. To-wttsend, Superintendent Public Schools, Hudson, N. Y. 
Every human being is appointed to take charge of his own body; and of all books 
written upon this subject, I know of none which will so well prepare one to do this as 
" Jarvis' Physiology" — that is, in so small a compass of matter. It considers the 
pure, simple laws of health paramount to science ; and though the work is thoroughly 
scientific, it is divested of all cumbrous technicalities, and presents the subject of phy- 
sical life in a manner and style really charming. It is unquestionably the best text- 
book on physiology I have ever seen. It is giving great satisfaction in the schools oi 
this city, where it has been adopted as the standard. 

From L. J. Saiifoed, M.D., Prof. Anatomy and Physiology in Yale College 
Books on human physiology, designed for the use of schools, are more generally a 
failure perhaps than are school-books on most other subjects. 

The great want in this department is met, we think, in the well-written treatise o' 
Dr. Jarvis, entitled " Physiology and Laws of Health." * * The work is not too 
detailed nor too expansive in any department, and is clear and concise in all. It ia 
not burdened with an excess of anatomical description, nor rendered discursive by 
■many zoological references. Anatomical statements are made to the extent of quali- 
fying the student to attend, under standingly, to an exposition of those functional pro 
cesses which, collectively, make up health; thus the laws of health are enunciated, 
and many suggestions are given which, if heeded, will tend to its preservation. 



For further testimony of similar character, see current numbers of the lllua 
irated F^ducational Bulletin. 

33 



T?ie National Series of Standard School-Hooks. 



NATURAL SCIENCE. 

"FOUKTEEN WEEKS" IN EACH BEMCH, 

By J. DORMAN STEELE, A. M. 

Steele's 14 Weeks Course in Chemistry g? $1 50 
Steele's 14 Weeks Course in Astronomy . i 53 
Steele's 14 Weeks Course in Philosophy • X so 
Steele's 14 Weeks Course in Geology. . 1 $® 
Steele's 14 Weeks Course in Physiology ■ 1 50 

Our Text-Books in these studies are, as a general thing, dull and uninteresting. 
They contain from 400 to 600 pages of dry facts and unconnected details. They 
abound in that which the student cannot learn, much less remember. The pupil 
commences the study, is confused by the fine print and coarse print, and neither 
knowing exactly what to learn nor what to hasten over, is crowded through the 
single term generally assigned to each branch, and frequently comes to the close 
without a definite and exact idea of a single scientific principle. 

Steele's Fourteen Weeks Courses contain only that which eveiy well-informed 
person should know, while all that which concerns only the professional scientist 
is omitted. The language is clear, simple, and interesting, and the illustrations 
bring the subject within the range of home life and daily experience. They give 
such of the general principles and the prominent facts as a pupil can make famil- 
iar as household words within a single term. The type is large and open ; there 
is no fine print to annoy ; the cuts are copies of genuine experiments or natural 
phenomena, and are of fine execution. 

In fine, by a system of condensation peculiarly his own, the author reduces each 
branch to the limits of a single term of study, while sacrificing nothing that is es- 
sential, and nothing that is usually retained from the study of the larger manuals 
in common use. Thus the student has rare opportunity to economize his time, or 
rather to employ that which he has to the best advantage. 

A notable feature is the author's charming " style," fortified by an enthusiasm 
over his subject in which the student will not fail to partake. Believing that 
Natural Science is full of fascination, he has moulded it into a form that attracts 
the attention and kindles the enthusiasm of the pupil. 

The recent editions contain the author's "Practical Questions" on a plan never 
before attempted in scientific text-books. These are questions as to the nature 
and cause of common phenomena, and are not directly answered in the text, the 
design being to test and promote an intelligent use of the student's knowledge of 
the foregoing principles. 

Steele's General Key to his Works- . • • *i 50 

This work is mainly composed of Answers to the Practical Questions and Solu- 
tions of the Problems in the author's celebrated "Fourteen Weeks Courses " in 
the several sciences, with many hints to teachers, minor Tables, &c. Should b« 
on every teacher's desk. 

34 



The National Series of Standard School-^Books* 

fc_ — _ — — _ , _ „ « 

Steele's 14 Weeks in each Science. 

TESTIMONIALS. 

From L. A. Bikle, President N. C. College. 
I have not been disappointed. Shall take pleasure in introducing this series. 

From J. F. Cox, Prest. Southern Female College, Ga. 
1 am much pleased with these books, and expect to introduce them. 

From J. R. Branham, Prin. Brownsville Female College, Tenn. 
They are capital little books, and are now in use in our institution. 

From W. H. Goodale, Professor Beadville Seminary, La. 
We are using your 14 Weeks Course, and are much pleased with them. 

From W. A. Boles, Supt. Shelbyville Graded School, Ind. 
They are as entertaining as a story book, and much more improving to the mind. 

From S. A. Snow, Principal of High School, Uxbridge, Mass. 

Steele's 14 Weeks Courses in the Sciences are a perfect success. 

From John W. Doughty, Newburg Free Academy, N. Y. 

T was prepared to find Prof. Steele's Course both attractive and instructive. My 
highest expectations have been fully realized. 

From J. S. Blackwell, Prest. Ghent College, Ky. 

Prof. Steele's unexampled success in providing for the wants of academic classes, 
has led me to look forward with high anticipations to his forthcoming issue. 

From J. F. Cook, Prest. La Grange College, Mo. 

I am pleased with the neatness of these books and the delightful diction. I have 
been teaching for years, and have never seen a lovelier little volume than the As- 
tronomy. 

From M. W. Smith, Prin. of High School, Morrison, HI. 

They seem to me to be admirably adapted to the wants of a public school, con- 
taining, as they do, a sufficiently comprehensive arrangement of elementary prin- 
ciples to excite a healthy thirst for a more thorough knowledge of those sciences. 

From J. D. Barteey, Prin. of High School, Concord, N. H. 

They are just such books as I have looked for, viz., those of interesting style, 
not cumbersome and filled up with things to be omitted by the pupil, and yet suf- 
ficiently full of facts for the purpose of most scholars in these sciences in our high 
schools ; there is nothing but what a pupil of average ability can thoroughly 
master. 

From Alonzo Norton Lewis, Principal of Parker Academy, Conn. 

I consider Steele's Fourteen Weeks Courses in Philosophy, Chemistry, &c, the 
best school-books that have been issued in this country. 

As an introduction to the various branches of which they treat, and especially 
for that numerous class of pupils who have not the time for a more extended 
course, I consider them invaluable. 

From Edward Brooks, Prin. State Normal School, MUlersville, Pa. 

At the meeting of Normal School Principals, I presented the following resolu- 
tion, which was unanimously adopted: '''•Resolved, That Steele's 14 Weeks 
Courses in Natural Philosophy and Astronomy, or an amount equivalent to what 
is contained in them, be adopted for use in the State Normal Schools of Pennsyl- 
vania." The works themselves will be adopted by at least three of the schools, 
and, I presume, by them all. 

35 



National Series of Standard Sc?iool-tBooks. 

LITERATURE." 

^ ^ » 

Cleveland's Compendiums .... each, $*2 50 

English Literature. American Literature. 

English Literature of the XIXth Century. 

In these volumes are gathered the crearu of the literature of the English speak, 
ing people for the school-room and the general reader. Their reputation is 
national. More than 125,000 copies have been sold. 

Boyd's English Classics each, *i 25 

Milton's Paradise Lost. Thomson's Seasons. 

Young's Night Thoughts. Pollok's Course op Time. 

Cowper's Task, Table Talk, &c. Lord Bacon's Essays. 

This series of annotated editions of great English writers, in prose and poetry, 
is designed for critical reading and parsing in schools. Prof. J. K. Boyd proves 
himself an editor of high capacity, and the works themselves need no encomium. 
As auxiliary to the study of Belles Lettres, etc., these works have no equal. 

Pope's Essay on Man . *20 

Pope's Homer's Iliad *80 

The metrical translation of the great poet of antiquity, and the matchless 
'• Essay on the Nature and State of Man," by Alexander Pope, afford superior 
exercise in literature and parsing. 

AESTHETICS. 



Huntington's Manual of the Fine Arts • -*i 75 

A view of the rise and progress of Art in different countries, a brief 
account of the most eminent masters of Art, and an analysis of the prin- 
ciples of Art. It is complete in itself, or may precede to advantage the 
critical work of Lord Kames. 

Boyd's Kames' Elements of Criticism . .*i 75 

The best edition of this standard work ; without the study of which 
none may be considered proficient in the science of the Perceptions. No 
other study can be pursued with so marked an effect upon the taste and 
refinement of the pupil. 

POLITICAL ECONOMY. 



Champlin's Lessons on Political Economy l 25 

An improvement on previous treatises, being shorter, yet containing 
«very thing essential, with a view of recent questions in finance, etc, 
which is not elsewhere found. 

36 



The JVationag Series of Standard School-^Books. 

' — . 

CLEVELAND'S COMPENDITJMS. 



TESTI3SdI03SriA.XjS. 

From the New Englander. 
This is the very hest book of the kind we have ever examined. 

From George B. Emerson, Esq., Boston. 
The Biographical Sketches are just and discriminating ; the selections are admir- 
able, and I have adopted the work as a text-book for my first class. 

From Prof. Moses Coit Tyler, of the Michigan University. 
I have given your book a thorough examination, and am greatly delighted with 
It ; and shall have great pleasure in directing the attention of my classes to a woris 
which affords so admirable a bird's-eye view of recent "English Literature." 
From the' Saturday Review. 
It acquaints the reader with the characteristic method, tone, and quality of all the 
chief notabilities of the period, and will give the careful student a better idea of the 
recent history of English Literature than nine educated Englishmen in ten possess. 
From the Methodist Quarterly Beview, New York. 
This work is a transcript of the best American mind; a vehicle of the noblest 
American spirit. No parent who would introduce his child to a kuowledge of our 
country's literature, and at the same time indoctrinate his heart in the purest prin- 
ciples, need fear to put this manual in the youthful hand. 

From Kev. C. Peirce, Principal, West Newton, Mass. 
I do not believe the work is to be found from which, within the same limits, so 
much interesting and valuable information in regard to English writers and English 
literature of every age, can be obtained ; and it deserves to find a place in all our 
high schools and academies, as well as in every private library. 

From the Independent. 
The work of selection and compilation — requiring a perfect familiarity with the 
whole range of English literature, a judgment clear and impartial, a taste at once 
delicate and severe, and a most sensitive regard to purity of thought or feeling — has 
been better accomplished in this than in any kindred volume with which we are 
acquainted. 

From the Christian Examiner. 
To form such a Compendium, good taste, fine scholarship familiar acquaintance 
with English literature, unwearied industry, tact acquired by practice, an interest 
in the culture of the young, a regard for truth, purity, philanthropy, religion, as the 
highest attainment and the highest beauty, — all these were needed, and they are 
united in Mr. Cleveland. 

CHAMPLIN'S POLITICAL ECONOMY. 



From J. L. Bothwele, Prin. Public School No. lh, Albany, N. Y. 

I have examined Champlin's Political Economy with much pleasure, and shall be 
pleased to put it into the hands of my pupils. In quantity and quality I think it 
superior to anything that I have examined. 

From Pres. N. E. Cobleigh, East Tennessee Wesleyan University. 

An examination of Champlin's Political Economy has satisfied me that it is the 
book I want. For brevity and compactness, division of the subject, and clear state- 
ment, and for appropriateness of treatment, I consider it a better text-book than 
any other in the market. 

From the Evening Mail, New York. 

A new interest has been imparted to the science of political economy since we 
have been necessitated to raise such vast sums of money for the support of the gov- 
ernment. The time, therefore, is favorable for the introduction of works like the 
above. This little volume of two hundred pages is intended for beginners, for the 
common school and academy. It is intended as a basis upon which to rear a more 
elaborate superstructure. There is nothing in the principles of political economy 
above the comprehension of average scholars, when they are clearly set forth. This 
eeems to have been done by President Champlin in an easy and graceful manner. 

37 



The National Series of Standard School-'Boolcs, 

ELOCUTION. 



Taverner Graham's Reasonable Elocution, U 25 

Based upon the belief that true Elocution is the right interpreta- 
tion of Thought, and guiding the student to an intelligent appre- 
ciation, instead of a merely mechanical knowledge, of its rules. 

Zachos' Analytic Elocution 1 50 

All departments of elocution — such as the analysis of the voice and the 

sentence, phonology, rhythm, expression, gesture, &c are here arranged 

for instruction in classes, illustrated by copious examples. 

Sherwood's Self Culture • 1 00 

Self-culture in reading, speaking, and conversation — a very valuable 
treatise to those who would perfect themselves in these accomplishments. 

SPEAKERS. 

Northend's Little Orator, *60-Child's Speaker*60 

Two little works of the same grade but different selections, containing 
simple and attractive pieces for children under twelve years of age. 

Northend's Young Declaimer *75 

Northend's National Orator *i 25 

Two volumes of Prose, Poetry, and Dialogue, adapted to inter- 
mediate and grammar classes respectively. 

Northend's Entertaining Dialogues • • - -*i 25 

Extracts eminently adapted to cultivate the dramatic faculties, as well 
as entertain an audience. 

Swett's Common School Speaker . . . .*l 25 

Selections from recent literature. 

Raymond's Patriotic Speaker . . . . . .*2 00 

A su per d compilation ot modern eloquence and poetry, with original 
dramatic exercises. Nearly every eminent living orator is represented, 
without distinction of place or party. 

COMPOSITION, &cT 

Brookfield's First Book in Composition • 50 

Making the cultivation of this important art feasible for the smallest 
child. By a new method, to iuduce and stimulate thought 

Boyd's Composition and Rhetoric • - • • 1 50 

This work furnishes all the aid that is needful or can be desired in 
the various departments and styles of composition, both inprcse and verse. 

Day's Art of Rhetoric 1 25 

Noted for exactness of definition^ clear limitation, and philosophical 
development of subject ; the large share of attention given to InventWn, 
as a branch of Rhetoric, and the unequalled analysis of style 

38 



T7ie JVatio?ial Series of Standard School-ffooks* 

MENTAL PHILOSOPHY. 



^ ■ ♦ •<*- 



Mahan's Intellectual Philosophy *i 75 

The subject exhaustively considered. The author has evinced learning, candor, 
and independent thinking. 

Mahan's Science of Logic % 00 

A profound analysis of the laws of thought. The system possesses the merit of 
being intelligible and self consistent. In addition to the author's carefully elabo- 
rated views, it embraces results attained by the ablest minds of Great Britain, Ger- 
many, and France, in this department. 

Boyd's Elements of Logic i 25 

A systematic and philosophic condensation of the subject, fortified with additions 
from Watts, Abercrombie, Whately, &c. 

Watts on the Mind • so 

The Improvement of the Mind, by Isaac Watts, is designed as a guide for the 
attainment of useful knowledge. As a text-book it is unparalleled; and the disci- 
pline it affords cannot be too highly esteemed by the educator. 



MORALS. 



■»«♦■». 



Peabody's Moral Philosophy 1 25 

A sho^t course ; by the Professor of Christian Morals, Harvard University — for 
the Freshman Class and for High Schools. 

Alden's Text-Book of Ethics 60 

For young pupils. To aid in systematizing the ethical teachings of the Bible, 
and point out the coincidences between the instructions of the sacred volume and 
the sound conclusions of reason. 

Willard's Morals for the Young 75 

Lessons in conversational style to inculcate the elements of moral philosophy. 
The study is made attractive by narratives and engravings. 

GOVERNMENT. 

Howe's Young Citizen's Catechism .... 75 

Explaining the duties of District, Town, City, County, State, and United States 
Officers, with rules for parliamentary and commercial business — that which every 
future " sovereign " ought to know, and so few are taught. 

Young's Lessons in Civil Government . . l 25 

A comprehensive view of Government, and abstract of the laws showing the 
rights, duties, and responsibilities of citizens. 

Mansfield's Political Manual l 25 

This is a complete view of the theory and practice of the General and State Gov- 
ernments of the United States, designed as a text-book. The author is an esteemed 
and able professor of constitutional law, widely known for his sagacious utterances 
in matters of statecraft through the public press. Recent events teach with em- 
phasis the vital necessity that the rising generation should comprehend the noble 
polity of the American government, that they may act intelligently when endowed 
with a voice in it. 

39 



Thk National Series of Standard School-T>ooks, 

MODERN LANGUAGE. 



French and English Primer, $10 

German and English Primer, 10 

Spanish and English Primer, 30 

The names of common objects properly illustrated and arranged in easy 
lessons. 

Ledru's French Fables, 75 

Ledru's French Grammar, ....... 1 00 

Ledru's French Reader, . . . * l 00 

The author's long experience has enabled him to present the most thor- 
oughly practical text-books extant, in this branch. The system \>f pro- 
nunciation (by phonetic illustration) is original with this author, and will 
commend itself to all American teachers, as it enables their pupils to se- 
cure an absolutely correct pronunciation without the assistance of a native 
master. This feature is peculiarly valuable also to " self-taught" students. 
The directions for ascertaining the gender of French nouns — also a great 
Gtumbling-block — are peculiar to this work, and will be found remarkably 
competent to the end proposed. The criticism of teachers and the test of 
the school-room is invited to this excellent series, with confidence. 

Worman's French Echo, 1 25 

To teach conversational French by actual practice, on an entirely new 
plan, which recognizes the importance of the student learning to think in 
the language which he speaks. It furnishes an extensive vocabulary of 
words and expressions iH common use, and suffices to free the learner 
from the embarrassments which the peculiarities of his own tongue are 
likely to be to him, and to make him thoroughly familiar with the use 
of proper idioms. 

Worman's German Echo, ........ 1 2/1 

On the same plan. See Worman's German Series, page 42. 

Pujol's Complete French Class-Book, ... 2 25 

Offers, in one volume, methodically arranged, a complete French course 
—usually embraced in series of from five to twelve books, including the 
bulky and expensive Lexicon. Here are Grammar, Conversation, and 
choice Literature — selected from the best French authors. Each branch 
is thoroughly handled ; and. the student, having diligently completed the 
course as prescribed, may consider himself, without further application, 
au/ait in the most polite and elegant language of modern times. 

Maurice-Poitevin's Grammaire Francaise, • 1 00 

American schools are at last supplied with an American edition of this 
famous text-book. Many of our best institutions have for years been pro- 
curing it from abroad rather than forego the advantages it oifers. The 
policy of putting students who have acquired some proficiency from the 
ordinary text-books, into a Grammar written in the vernacular, can not 
l)e too highly commended. It affords an opportunity for finish and. review 
at once ; while embodying abundant practice of its own rules. 

Joynes' French Pronunciation, so 

Willard's Historia de los Estados Unidos, - 2 00 

The History of the United States, translated by Professors Tolon and 
Db Tornos, will be found a valuable, instructive, and entertaining read- 
ing-book for Spanish classes. A „ 



The National Series of Standard School- Books. 

Pujol's Complete French Olass-Book. 

TESTIMONIALS. 

From Prof. Elias Peissnek, Union College. 
I take great pleasure in recommending Pujol and Van Norman's French Class- 
Book, as there is no French grammar or class-book which, can be compared with 
it in completeness, system, clearness, and general utility. 

F*"om Edward North, President of Hamilton College. 
I have cavpfnily examined Pujol and Van Norman's French Class-Book, and am 
satisfied of its superiority, for college purposes, over auy other heretofore used; 
*Ve shali p'jt fail to use it with our next class in French. 

Fr^rr A. Curtis, Preset of Cincinnati Literary and Scientific Institute. 
I am confident that it may he made an instrument in conveying to the student, 
hi fr jm six months to a year, the art of speaking and writing the French with 
-almost native fluency and propriety. 

From Hiram Orcutt, A. M., Prin. Glenwood and Tilden Ladies' 1 Seminaries. 

I have used Pujol's French Grammar in my two seminaries, exclusively, for 
2iore than a year, and have no hesitation in saying that I regard it the best text- 
book in this department extant. And my opinion is confirmed by the testimony 
of Prof. F. De Launay and Mademoiselle Marindin. They assure me that the 
book is eminently accurate and practical, as tested in the school-room. 

From Prof. Theo. F. De Fumat, Hebrew Educational Institute, Memphis, Term. 
M. Pujol's French Grammar is one of the best and most practical works. The 
French language is chosen and elegant in style — modern and easy. It is far su- 
perior to the other French class-bocks in this country. The selection of the con- 
versational part is very good, and will interest pupils ; and being all ccmpleted in 
only one volume, it is especially desirable to have it introducedln our schools. 

From Prof. Ja^tes H. Worman, Bordentown Female College, JV. J. 
The work is upon the same plan as the text-books for the study of French and 
English published in Berlin, for the study of those who have not the aid of a 
teacher, and these books are considered, by the first authorities, the best books. 
In most of our institutions, Americans teach the modern languages, and hereto- 
fore the trouble has been to give them a text-book that would dispose of the 
difficulties of the French pronunciation. This difficulty is successfully removed 
by P. and Van N., and I have every reason to believe it will soon make* its way 
into most of our best schools. 

From Prof. Charles S. Pod, Ann Smith Academy, Lexington, Va. 
I cannot do better than to recommend " Pujol and Van Norman." For compre- 
hensive and systematic arrangement, progressive and thorough development of 
all grammatical principles and idioms, with a due admixture of theoretical knowl- 
edge and practical exercise, I regard it as superior to any (other) book of the kind. 

From A. A. Forster, Prin. Pinehurst School, Toronto, C. W. 
I have great satisfaction in bearing testimony to M. Pujol's System of French 
Instruction, as given in his complete class-book. For clearness and comprehen- 
siveness, adapted for all classes of pupils, I have found it superior to any other 
vrork of the kind, and have now used it for some years in my establishment with 
great success. 

From Prof. Otto Fedder, Maplewood Institute, Pittsfleld, Mass. 
The conversational exercises will prove an immense saving of the hardest kind 
ot labor to teachers. There is scarcely any thing more trying in the way of 
teaching language, than to rack your brain for short and easily intelligible bits 
of conversation, and to repeat them time and again with no better result than 
extorting at long intervals a doubting " oui," or a hesitating " Don, monsieur " 



$W For further testimony of a similar character, see special circular, sod 
current numbers of the Educational Bulletin. 

41 



The National Series of Standard School- Books. 

GERMAN. 



A COMPLETE COUKSE IN THE GEEMAU. 

By JAMES H. WORMAN, A. M. 

Worman's Elementary German Grammar .11 50 
Worman's Complete German Grammar ■ 2 00 

\ f These volumes are designed for intermediate and advanced classes respectively.. 
Though following the same general method with "Otto" (that of 'Gaspey'). 
our author differs essentially in its application. He* is more practical, more sys- 
tematic, more accurate, and besides introduces a number of invaluable features 
which have never before been combined in a German grammar. 

Among other things, it may be claimed for Prof. Worman that he has been 
the first to introduce in an American text-book for learning German, a system 
of analogy and comparison with other languages. Our best teachers are also 
enthusiastic about his methods of inculcating the art of speaking, of understanding 
the spoken language, of correct pronunciation ; the sensible and convenient origi' 
nal classification of nouns (in four declensions), and of irregular verbs, also de- 
serves much praise. We also note the use of heavy type to indicate etymological 
changes in the paradigms, »nd, in the exercises, the parts which specially illustrate 
preceding rules. 

Worman's Elementary German Reader . . 1 25 
Worman's Collegiate German Reader ... 2 00 

The finest and most judicious compilation of classical and standard German 
Literature. These works embrace, progressively arranged, selections from the 
masterpieces of Goethe, Schiller, Korner, Seume, Uhland, Freiligrath, Heine, 
Schlegel, Holty, Lenau, "Wi eland, Herder, Lessing, Kant, Fichte, Schelling, Win- 
kelmann, Humboldt, Eanke, Raumer, Menzel, Gervinus, &c, and contains com- 
plete Goethe's "IpLigenie," Schiller's " Jungfrau;" also, for instruction in mod- 
ern conversational German, Benedix's " Eigensinn." 

There are besides, Biographical Sketches of each author contributing, Notes, 
explanatory and philological (after the text), Grammatical Eeferences to all lead- 
ing grammars, as well as the editor's own, and an adequate Vocabulary. 

Worman's German Echo l 25 

Consists of exercises in colloquial style entirely in the German, with an ade- 
quate vocabulary, not only of words but of idioms. The object of the system de- 
veloped in this work (and its companion volume in the French) is to break up the 
laborious and tedious habit of translating the thoughts, which is the student's 
most effectual bar to fluent conversation, and to lead him to think in the language 
in which he speaks. As the exercises illustrate scenes in actual life, a considera- 
ble knowledge of the manners and customs of the German people is also acquired 
from the use of this manual. . 

Worman's German Copy-Books, 3 Numbers, each 15 

On the same plan as the most approved systems for English penmanship, with 
progressive copies. 

42 



The National Series of Sta?idard School- Hooks* 

Worman's German Grammars. 

TESTIMONIALS. 

From, Prof. It. "W. Jones, Petersburg Female College, Va. 
From what I have seen of the work it is almost certain / shall introduce it into 
this institution. 

From Prof. G. Campbell, University of Minnesota. 
A valuable addition to our school-books, and will find many friends, and do great 
good. 

From Prof. O. II P. Corpeew, Mary Military Inst , Md. 

I am better pleased with them than any I have ever taught. I have already ordered 
through our booksellers. 

From Prof. E. S. Kendall, Vernon Academy, Conn. 
I at once put the Elementary Grammar into the hands of a class of beginners, and 
have used it with great satisfaction. 

From Prof. D. E. Holmes, Berlin Academy, Wis. 
"Worman's German works are superior. I shall use them hereafter in my German 
classes. 

From Prof. Magnus Buciiiioltz, Hiram, College, Ohio. 

I have examined the Complete Grammar, and find it excellent. You may rely that 
it will be used here. 

From Prin. Thos. "W. Tobev, Paducah Female Seminary, Ky. 
The Complete German Grammar is worthy of an extensive circulation. It is ad' 
mirably adapted to the class-room. I shall use it. 

From, Prof. Alex. Eosenspitz, Houston Academy, Texas. 
Bearer will take and pay for 3 dozen copies. Mr. Worman deserves the approbation 
and esteem of the teacher and the thanks of the student. 

From Prof. G. Malmene, Augusta Seminary, Maine. 
t The Complete Grammar cannot fail to give great satisfaction by the simplicity 
of its arrangement, and by its completeness. 

From Prin. Oval Pikket, Christian University, Mo. 
Just such a series as is positively necessary. I do hope the author will succeed as 
well in the French, &c, as he has in the German. 

From Prof. S. D. Hillman, Dickinson College, Pa. 
The class have lately commenced, and my examination thus far warrants me in say- 
ing that I regard it as the best grammar for instruction in the German. 

From Prin. Silas Liveemoue, Bloomfield Seminary, Mo. 
I have found a classically and scientifically educated Prussian gentleman whom I 
propose to make German instructor. I have shown him both your German grammars. 
He has expressed his approbation of them generally. 

From Prof. Z. Test, Rowland School for Young Ladies, JSF. T. 
I shall introduce the books. From a cursory examination I have no hesitation in 
pronouncing the Complete Grammar a decided improvement on the text-books at 
present in use in this country. 

From Prof. Lewis Kistler, Northwestern University, III. 
Having looked through the Complete Grammar with some car^ I must say that you 
have produced a good book ; you may be awarded with this gratification — that your 
grammar promotes the facility of learning the German language, and of becoming 
acquainted with its rich literature. 

From Pres. J. P. Eous, Stockwell Collegiate Inst., Ind. 
I supplied a class with the Elementary Grammar, and it gives complete satisfac- 
Hon. The conversational and reading exercises are well calculated to illustrate the 
principles, and lead the student on an easy yet thoreugh course. I think the Com 
plete Grammar equally attractive. 

43 



National Series of Standard Sc/iool-^oo&s. 



THE CLASSICS. 

» » » = 

LATIN. 
Silber's Latin Course, •' $1 25 

The book contains an Epitome of Latin Grammar, followed by Reading Exercises 
with explanatory Notes and copious References to the leading Latin Grammars and 
also to the Epitome which precedes the work. Then follow a Latin-English Vocabu- 
lary and Exercises in Latin Prose Composition, being thus complete in itself and a 
very suitable work to put in the hands of one about to study the language. ' 

Searing's Virgil's /Eneid, 2 25 

It contains only the first six books of the iEneid. 2. A very carefully constructed 
Dictionary. 3. Sufficiently copious Notes. 4. Grammatical references to four lead- 
ing Grammars. 5. Numerous Illustrations of the highest order, 6. A superb Map 
of the Mediterranean and adjacent countries. 7. Dr. S. H. Taylor's "Questions on 
the ^Eneid." 8. A Metrical Index, and an Essay on the Poetical Style. 9, A photo- 
graphic foe simile of a>i early Latin M.S. 10. The text according to Jahn, but para- 
graphed according to Ladewig. 11. Superior mechanical execution. 

Blair's Latin Pronunciation, 1 oo 

An inquiry into the proper sounds of the Language during the Classical Period. 
By Prof. Blair, of Hampden Sidney College, Va. 



Andrews & Stoddard's Latin Grammar, *i so 

Andrews' Questions on the Grammar, • *o 15 

Andrews' Latin Exercises, *i 25 

Andrews' Yiri Romae, *i 25 

Andrews' Sallust's Jugurthine War, &c. *i so 

Andrews' Eclogues & Georgics of Virgil, *i 50 

Andrews' Caesar's Commentaries, . . . . *i 50 

Andrews' Ovid's Metamorphoses, • • • *i 25 

GREEK. 

Crosby's Greek Grammar, 2 oo 

Crosby's Xenophon's Anabasis, i 25 

Searing's Homer's Iliad, 

MYTHOLOGY. 
Dwight's Grecian and Roman Mythology. 

School edition, $1 25; ^University edition, *3 00 

A knowledge of the fables of antiqaity, thus presented in a systematic form, is as 
indispensable to the student of general literature as to him who would peruse intelli- 
gently the classical authors. The mythological allusions eo frequent in literature are 
readily understood with such a Key as this. 

44 



'The National Se?'ies of Standard Sc7iool-3Books. 

SEARING'S VIRGIL. 



SPECIMEN FRAGMENTS OP LETTERS. 

"I adopt it gladly."— Pres. Y. Dabney, Loudoun School, Va. 

"Hike Searing's Virgil."— Prof. Bristol, Eipon College, Wis. 

'•Meets my desires very thoroughly."— Prof. Clark, Berea College, Ohio. 

" Superior to any other edition of Virgil." — Pres. Hall, Macon College, Mo. 

"Shall adopt it at once."— Pres. B. P. Baker, Searcy Female Institute, Ark. 

" Your Virgil is a beauty.''''— Prop. W. H. De Motte, Illinois Female College. 

" After use, I regard it the best."— Pres. G. H. Barton, Rome Academy, N. T. 

"We like it better every day." — Pres. R. K. Bpehrle, Allentown Academy, Pa. 

" I am delighted with your Virgil."— Pre*. W. T. Leonard, Pierce Academy, Mass. 

" Stands well the test of class-room."— Pres. P. A. Chase, Lyons Col. Inst., Iowa. 

"I do not see how it can be improved."— Pees. N. F. D. Browne, Charl. Hall, Md. 

"The most complete that I have seen." — Pres. A. Brown, Columbus High School, 
Ohio. 

" Our Professor of Language very highly approves." — Sept. J. G. James, Texas 
Military Institute. 

"It responds to a want long felt by teachers. It is beautiful and complete." — 
Prof. Brooks, University of Minnesota. 

" The ideal edition. We want a few more classics of the same sort." — Pres. C. F. 
P. Bancroft, Lookout Mountain Institute, Tenn. 

" I certainly have never seen an edition so complete with important requisites for 
a student, nor with such fine text and general mechanical execution." — Pres. J. P. 
Park, University of Beseret, Utah. 

"It is charming both in its design and execution. And, on the whole, I think it 
*8 the best thing of the kind that I have seen." — Prof. J. De F. Richards, Pres. 
pro tern, of University of Alabama. 

" In beauty of execution, in judicious notes, and. in an adequate vocabulary, it 
merits all praise. I shall recommend its introduction." — Pres. J. K. Patterson, 
Kentucky Agricultural and Mechanical College. 

" Containing a good vocabulary and judicious notes, it will enable the industrious 
student to acquire an accurate knowledge of the most interesting part of Virgil's 
works." — Prof. J. T. Dunkles, Fant Alabama College. 

"It wants no element of completeness. It is by far the best classical text-book 
■with which I am acquainted. The notes are just right. They help the student 
when he most needs help." — Pres. C. A. Benker, Caledonia Grammar School, Vt. 

"I have examined Searing's Virgil with interest, and find that it more nearJy 
meets the wants of students than that of any other edition with which I am ac- 
quainted. I am able to introduce it to some extent at once."— Pres. J. Easter, 
Fast Genesee Conference Seminary. 

" I have been wishing to get a sight of it, and it exceeds my expectations. It is 
a beautiful book in every respect, and bears evidence of careful and critical study. 
The engravings add instruction as well as interest to the work. I shall recommend 
it to my classes."— Pres. Chas. H. Chandler, Glenwood Ladies' 1 Seminary. 

"A. S. Barnes & Co. have published an edition of the first six books of Virgil's 
yEneid. which is superior to its predecessors in several respects. The publishers 
have done a good service to the cause of classical education, and the book deserves 
a larcre circulation." — Prof. George W. Collord, Brooklyn Polytechnic, N. Y. 

v - My attention was called to Searing's Virgil by the fact of its containing a voca- 
bulary which would obviate the necessity of procuring a lexicon. But use in th4 
class-room has impressed me most favorably with the accuracy and just proportion 
of its notes, and the general excellence of its grammatical suggestions. The gen- 
eral character of the book in its paper, its typography, and its engravings is highly 
commendable, and the fac-simile manuscript is a valuable feature. I take great 
pleasure in commending the book to all who do not wish a complete edition of 
Virgil. It suits our short school courses admirably."— Henry L. Boltwood, Master 
qt Princeton High School, Iff. 

45 



The National Series of Standard Sc?iool-33ooks. 

RECORDS. 



Cole's Self-Reporting Class-Book, • . • . *$o 50 

For saving the Teacher's labor in averaging. At each opening are a full set ol 
Tables showing any scholar's standing at a glance and entirely obviating the neces- 
sity of computation. 

Tracy's School-Record, *o 75. Pocket edition, *o 65 

For keeping a simple but exact record of Attendance, Deportment, and Scholar- 
ship. The larger edition contains also a Calendar, an extensive list of Topics for 
Compositions and Colloquies, Themes for Short Lectures, Suggestions to Young 
Teachers, etc. 

Brooks' Teacher's Register, • *i oo 

Presents at one view a record of Attendance, Recitations, and Deportment for the 
whole term. 

Carter's Record and Roll-Book, *i so 

This is the most complete and convenient Record offered to the public. Besides 
the usual spaces for General Scholarship, Deportment, Attendance, etc., for each 
name and day, there is a space in red lines enclosing six minor spaces in blue for 
recording Recitations, 

National School Diary, ..... Per dozen, *i oo 

A little book of blank forms for weekly report of the standing of each scholar, 
from teacher to parent. A great convenience. 



REWARDS. 



National School Currency, .... Per set,*$i 50 

A little box containing certificates in the form of Money. The most entertaining 
and stimulating system of school rewards. The scholar is paid for his merits and 
fined for his shortcomings. Of course the most faithful are the most successful in 
business. In this way the use and value of money and the method of keeping 
accounts are also taught. One box of Currency will supply a school of fifty pupils. 



TACTICS. 



The Boy Soldier, 75 

Complete Infantrv Tactics for Schools, with illustrations, for the use of those who 
would introduce this pleasing relaxation from the confining duties of the desk. 

46 



The National Series of Standard SclioolSooks. 

CHAR 



McKenzie's Elocutionary Chart, $3 50 

Baade's Reading Case, *io oo 

This remarkable piece of school-room furniture is a receptacle containing a num- 
oer of primary cards. By an arrangement of slides on the front, one sentence at a 
cime is shown to the class. Twenty-eight thousand transpositions may he made, 
affording a variety of progressive exercises which no other piece of apparatus 
offers. One of its best features is, thatit is so exceedingly simple as not to get out 
cf order, while it may be operated with one ringer. 

Marcy's Eureka Tablet, *i so 

A new system for the Alphabet, by which it may be taught without fail in nine 
lessons. 

Scofield's School Tablets, *8 oo 

On Five Cards, exhibiting Ten Surfaces. These Tablets teach Orthography, 
Reading, Object-Lessons, Color, Form, etc. 

Watson's Phonetic Tablets, *8 oo 

Four Cards, and Eight Surfaces ; teaching Pronunciation and Elocution phonetic- 
any — for class exercises. 

Page's Normal Chart, *3 75 

The whole science of Elementary Sounds tabulated. By the author of Page's 
Theory and Practice of Teaching. 

Clark's Grammatical Chart, *3 75 

Exhibits the whole Science of Language in one comprehensive diagram. 

Davies' Mathematical Chart, *75 

Mathematics made simple to the eye. 

Monteith's Reference Maps (School Series), . .*20 oo 

Eight Numbers. Mounted on Pollers. Names all laid down in small type, so 
that to the pupil at a short distance they are Outline Maps, while they serve as 
their own key to the teacher. 

Willard's Chronographers, Each, *2 oo 

Historical. Four Numbers. Ancient Chronographer ; English Chronographer ; 
American Chronographer; Temple of Time (general). Dates and Events repre- 
sented to the eye. 

APPARATUS. 

Harrington's Geometrical Blocks, • • • .*$io oo 

These patented blocks are hinged, so that each form can be dissected. 

Harrington's Fractional Blocks, *8 00 

Steele's Chemical Apparatus, - . *20 oo 

Steele's Philosophical Apparatus, (see P .28) *125 oo 
Steele's Geological Cabinet, (see p.28) . . . *40 oo 
Wood's Botanical Apparatus, (see p.so ) . . *8 oo 
Bock's Physiological Apparatus^ .... 175 oo 

47 



The National Series of Standa?*d School-Sooks. 

MUSIC. 



-*++- 



Jepson's Music Readers. 3 vols. . . . Each, 75 eta 

These are not Looks from which children simply learn songs, par- 
rot-like, hut teach the subject progressively— the scholar learning^o 
read music by methods similar to those employed in teaching him to 
read printed language. Any teacher, however ignorant of music, pro- 
vided he can, upon trial, simply sound the scale, may teach it without 
assistance, and will end by being a good singer himself. The "Ele- 
mentary Music Reader," or first volume, heretofore issued hy another 
publisher, has attained results in the State of Connecticut, where only 
it has been known, entirely unprecedented in the history of teaching 
music. The two companion volumes carry the same method into the 
higher grades. 

Nash & Bristow's Cantara. No.i, $1.15; No. 2, 11.40 

The first volume is a complete musical text-hook for schools of 
every grade. No. 2 is a choice selection of Solos and Part Songs. 
The authors are Directors of Music in the public schools of New York 
City, in which these hooks are the standard of instruction. 

Curtis' Little Singer, so 60 

Curtis' School Vocalist, i oo 

Kingsley's School-Room Choir, 60 

Kingsley's Young Ladies' Harp, i oo 

Hager's Echo, 75 

Perkins' Sabbath Carols (for sunday-schoois), . . 35 

Phillips' Singing Annual do. do. . . 25 



DEVOTION. 



Brooks' School Manual of Devotion, • • • so 75 

This volume contains daily devotional exercises, consisting of a 
hymn, selections of Scripture for alternate reading by teacher and 
pupils, and a prayer. Its value for opening and closing school is 
apparent. i 



Brooks' School Harmonist, *?5 

ichh 

48 



Contains appropriate tunes for each hymn in the "Manual of Devo- 
tion" described above. 



The National Teachers' library. 



TEACHERS' LIBRARY. 



Object Lessons— Welch ........ .*$i oc 

This is a complete exposition of the popular modern system of 
" object teaching," for teachers of primary classes. 

Theory and Practice of Teaching— Page • • *i 50 

This volume hr-3, without doubt, been read by two hundred thousand 
teachers, and its popularity remains undiminished— large editions 
being exhausted yearly. It was the pioneer, as it is now the patri- 
arch of professional works for teachers. 

The Graded School-Wells *i 25 

The proper way to organize graded schools is here illustrated. The 
author has availed hisiself of the best elements of the several systems 
prevalent in Boston, New York, Philadelphia, Cincinnati, St. Louis, 
and other cities. ■ 

The Normal— Holbrook *i 50 

Carries a working school on its visit to teachers, showing the most 
approved methods of teaching all the common branches, including the 
technicalities, explanations, demonstrations, and definitions intro- 
ductory and peculiar to each branch. 

The Teachers' Institute— Fowle *i 25 

This is a volume of suggestions inspired by the author's experience 
at institutes, in the instruction of young teachers. A thousand points 
of interest to this class are most satisfactorily dealt with. 

Schools and Schoolmasters— Dickens • • • *i £3 

P ppropriate selections from the writings of the great novelist. 

The Metric System— Davie's *i so 

Considered with reference to its general introduction, and embrac- 
ing the views of John Quincy Adams and Sir John Herschel. 

The Student*— The Educator— Phelps - each,"! 50 
The Discipline of Life— Phelps , . , . ■ . 3i 76 

The authoress of these works is one of the most distinguished 
writers on education ; and they cannot fail to prove a valuable addi- 
tion to the School and Teachers 1 Libraries, being in a high degree 
botfi interesting and instructive. 

k Scientific Basis of Education— Hecker . .. *2 50 

Jj /rotation of study and classification by temperaments. 

49 



7%e National Teachers 9 Xibrary. 



American Education— Mansfield $1 50 

A treatise on the principles and elements of education, as practiced in 
this country, with ideas towards distinctive republican and Christian edu- 
cation. 

American Institutions— De Tocqueville ■ .*i 50 

A valuable index to the genius of our Government. 

Universal Education— Mayhew *i 75 

The subject is approached with the clear, keen perception of one who 
has observed its necessity, and realized its feasibility and expediency 
alike. The redeeming and elevating power of improved common schools 
constitutes the inspiration of the volume. 

Higher Christian Education— Dwight - • *i 50 

A treatise on the principles and spirit, the modes, directions, and ra- 
eults of all true teaching ; showing that right education should appeal to 
every element of enthusiasm in the teacher 1 ., aturo. 

Oral Training Lessons— Barnard . . . . *i oo 

The object of this very useful work is to furnish material for instruc- 
tors to impart orally to their classes, in branches not usually taught in 
common schools, embracing all departments of Natural Science and 
much general knowledge. 

Lectures on Natural History— Chadbourne * 75 

Affording many themes for cral instruction in this interesting science — 
especially in schools where it is not pursued as a class exercise. 

Outlines of Mathematical Science— Davies *i oo 

A manual suggesting the best methods of presenting mathematical in- 
struction on the part of the teacher, with that comprehensive view of the 
whole which is necessary to the intelligent treatment of a part, in science. 

Nature & Utility of Mathematics— Davies ■ -*i 50 

An elaborate and lucid exposition of the principles which lie at the 
foundation of pure mathematics, with a highly ingenious application of 
their results to the development of the essential idea of the different 
branches of the science. 

Mathematical Dictionary— Davies & Peck -*5 oo 

This cyclopaedia of mathematical science defines with completeness, 
precision, and accuracy, every technical term, thus constituting a popular 
treatise on each branch, and a general view of the whole subject. 

School Architecture-Barnard *2 25 

Attention is here called to the vital connection between a good school- 
house and a good school, with plans and specifications for securing th« 
former in the most economical and satisfaetory manner. 

50 



The National Teachers' Ijibrary. 



Liberal Education of Women— Orton . • **i 50 

Treats of " the demand and the method ;" being a compilation of the best and 
most advanced thought on this subject, by the leading writers and educators in 
England and America. Edited by a Professor in Yassar College. 

Education Abroad— Northrop *i so 

A thorough discussiou of the advantages and disadvantages of sending American 
children to Europe to be educated ; also, Papers on Legal Prevention of Illiteracy, 
Study and Health, Labor as an Educator, and other kindred subjects. By the Hon. 
Secretary of Education for Connecticut. 

The Teacher and the Parent— Northend ■ . *i so 

A treatise upon common-school education, designed to lead teachers to view their 
calling in its true light, and to stimulate them to fidelity. 

The Teachers' Assistant— Northend . . . . *i so 

A natural continuation of the author's previous work, more directly calculated for 
daily use in the administration of school discipline and instruction. 

School Government— Jewell *i so 

Full of advanced ideas on the subject which its title indicates. The criticisms 
upon current theories of punishment and schemes of administration have excited 
general attention and comment. 

Grammatical Diagrams— Jewell *i oo 

The diagram system of teaching grammar explained, defended, and improved. 
The curious in literature, the searcher for truth, those interested in new inventions, 
as well as the disciples of Prof. Clark, who would see their favorite theory fairly 
treated, all want this book. There are many who would like to be made familiar 
with this system before risking its use in a class. The opportunity is here afforded. 

The Complete Examiner— Stone *i ss 

Consists of a series of questions on every English branch of school and academic 
instruction, with reference to a given page or article of leading text-books where 
the answer may be found in full. Prepared to aid teachers in securing certificates, 
pupils in preparing for promotion, and teachers in selecting review questions. 

School Amusements— Root *i so 

To assist teachers in making the school interesting, with hints upon the manage- 
ment of the school-room. Eules for military and gymnastic exercises are included. 
Illustrated by diagrams. 

Institute Lectures— Bates #1 so 

These lectures, originally delivered before institutes, are based upon various 
topics in the departments of mental and moral culture. The volume is. calculated 
to prepare the will, awaken the inquiry, and stimulate the thought of the zealoua 
teacher. 



Method of Teachers' Institutes— Bates • ■ • *?s 

Sets forth the best method of conducting institutes, with a detailed account of the 
object, organization, plan of instruction, and true theory of education on which 
such instruction should be based. 

History and Progress of Education .... *i so 

in al 
the p 
1 cor 

51 



The systems of education prevailing in all nations and ages, the gradual advance 
to the present time, and the bearing of the past upon the present in this regard, are 
worthy of the careful investigation of all concerned in education. 



National School Library, 



THE SCHOOL LIBRARY. 

The two elements of instruction and entertainment were never more happily com- 
bined than in this collection of standard hooks. Children and adults alike will here 
find ample food for the mind, of the sort that is easily digested, while not degener- 
ating to the level of modern romance. 



LIBRARY OF LITERATURE. 
Milton's Paradise Lost. Boyd's Illustrated Ed., $1 GO 

Young's Night Thoughts . . . . do. . . 1 60 

Cowper's Task, Table Talk, &c. . do. . . 1 60 

Thomson's Seasons ...... do. . . 1 60 

Pollok's Course of Time . . . . do. . . 1 go 

These works, models of the hest and purest literature, are beautifully illustrated, 
and notes explain all douhtful meanings. 

Lord Bacon's Essays (Boyd's Edition) ... 1 go 

Another grand English classic, affording the highest example of purity in lan- 
guage and style. 

The Iliad Of Homer. Translated by Pope. . . 80 

Those who are unahle to read this greatest of ancient writers in the original, 
should not fail to avail themselves of this metrical version. 

Compendium of Eng. Literature— Cleveland, 2 50 
English Literature of XlXth Century do. 2 50 
Compendium of American Literature do. 2 50 

Nearly one hundred and fifty thousand volumes of Prof. Cleveland's inimitable 
compendiums have been sold. Taken together they present a complete view of 
literature. To the man who can afford but a few books these will supply the place 
of an extensive library. From commendations of the very highest authorities the 
following extracts will give some idea of the enthusiasm with which the works are 
regarded by scholars : 

With the Bible and your volumes one might leave libraries without very painful 
regret. — The work cannot be found from which in the same limits so much interest- 
ing and valuable information may be obtained. — Good taste, fine scholarship, 
familiar acquaintance with literature, unwearied industry, tact acquired by practice, 
an interest in the culture of the young, and regard for truth, purity, philanthropy 
and religion are united in Mr. Cleveland. — A judgment clear and impartial, a taste 
at once delicate and severe. — The biographies are just and discriminating. — An 
admirable bird's-eye view.— Acquaints the reader with the characteristic method, 
tone, and quality of each writer.— Succinct, carefully written, and wonderfully com- 
prehensive in detail, etc., etc. 

Milton's Poetical Works— Cleveland . . . 2 50 

This is the very best edition of the great Poet It includes a life of the author, 
notes, dissertations on each poem, a faultless text, anO is the only edition of MiJtou 
with a complete verbal Index. 

52 



National School Zibrary. 



LIBRARY OF HISTORY. 
History of Europe— Alison 12 50 

A reliable and standard work, which covers with clear, connected, 
and complete narrative, the eventful occurrences transpiring from 
A. D. 1789 to 1815, being mainly a history of the career of Napoleon 
Bonaparte. 

History of England— Berard f 75 

Combining a history of the social life of the English people with that 
of the civil and military transactions of the realm. 

History of Rome— Ricord • • i 60 

Possesses all the charm of an attractive romance. The fables -with 
which this history abounds are introduced in such away as net to deceive 
the inexperienced reader, while adding vastly to the interest of the. work 
and affording a pleasing index to the genius of the Roman peopK Illus- 
trated. 

The Republic of America— Willard • • . 2 25 
Universal History in Perspective— Willard 2 25 

From these two comparatively brief treatises the intelligent mind may 
obtain a comprehensive knowledge of the history of the world in both 
hemispheres. Mrs. Willard' s reputation aB an historian is wide as the 
land. Illustrated. 

Ecclesiastical History— Marsh 2 oo 

A history of the Church in all ages, with a comprehensive review of all 
forms of religion fivm the creation of the world. No otner source affords, 
in the same compass, the information here conveyed. 

History of the Ancient Hebrews— Mills . . l 75 

The record of " God's people" from the call of Abraham to the destruc- 
tion of Jerusalem ; gathered from sources sacred and profane. 

The Mexican War— Mansfield i 50 

A history of its origin, and a detailed account of its victories ; with 
official despatches, the treaty of peace, and valuable tables. Illustrated. 



Early History of Michigan— Sheldon • • . 2 50 

A work of value and deep interest to the people of the West. Com- 
piled under the supervision of Hon. Lewis Cass. Portraits. 

History of Texas— Baker ....... .125 

A pithy and interesting resume. Copiously illustrated. The State 
constitution and extracts from the speeches and writings of eminent 
Texans are appended. 

53 



National School Zibrary, 



LIBRARY OF BIOGRAPHY. 
Life of Dr. Sam. Johnson— Boswell • • $2 25 

This work has been before the public for seventy years, with increasing 
approbation. Boswell is known as " the prince of biographers." 

Henry Clay's Life and Speeches— Mallory 

2 vols . - 4 50 

This great American statesman commands the admiration, and Ida 
character and deeds solicit the study of every patriot. 

Life & Services of General Scott— Mansfield l 75 

The hero of the Mexican war, who was for many years the most promi- 
nent figure in American military circles, should not be forgotten in the 
whirl of more recent events than those by which he signalized himself. 
Illustrated. 

Garibaldi's Autobiography l 50 

The Italian patriot's record of his own life, translated and edited by his 
friend and admirer. A thrilling narrative of a romantic career. With 
portrait. 

Lives of the Signers— Dwight l 50 

The memory of the noble men who declared our country free at the 
peril of their own "lives, fortunes, and sacred honor," should be em- 
balmed in every American 1 s heart. 

Life of Sir Joshua Reynolds— Cunningham i 50 

A candid, truthful, and appreciative memoir of the great painter, with 
a compilation of his discourses. The volume is a text-book for artists, as 
well as those who would acquire the rudiments of art With a portrait. 

Prison Life 75 

Interesting biographies of celebrated prisjners and martyrs, designed 
especially for the instruction and cultivation of youth. 

54 



National School library. 



LIBRARY OF NATURAL SCIENCE. 
The Treasury of Knowledge ti 2a 

A cyclopaedia of ten thousand common things, embracing the widest 
range of subject-matter. Illustrated. 

Canot's Popular Physics l 75 

The elements of natural philosophy for both student and the general 
reader. The original work is celebrated for the magnificent character of 
its illustrations, all of which are literally reproduced here. 

Principles of Chemistry— Porter 2 oo 

A work which commends itself to tne amateur in science by its extreme 
simplicity, and careful avoidance of unnecessary detail. Illustrated. 

Class-Book of Botany— Wood 3 50 

Indispensable as a work of reference. Illustrated. 

The Laws of Health— Jarvis 1 65 

This is not an abstract anatomy, but all its teachings are directed to the 
best methods of preserving health, as inculcated by an intelligent know- 
ledge of the structure and needs ef the human body. Illustrated. 

Vegetable & Animal Physiology— Hamilton l 25 

An exhaustive analysis of the conditions of life in all animate nature. 
Hiustrated. 

Elements of Zoology— Chambers 1 50 

A complete view of the animal kingdom as a poraon ef external nature. 
Illustrated. 

Astronography— Willard * 00 

The elements of astronomy in a compact and readable form. Illus- 
trated. 

Elements of Geology— Page ..... 1 25 

The subject presented in its two aspects of interesting and impertaat. 
Illustrated. 

Lectures on Natural History— Chadbourne 75 

The subject is here considered in its relations to intellect, taste, health, 
and religion. „ „ 

05 



National School Zibi*ary. 



LIBRARY OF TRAVEL. 
Life in the Sandwich Islands— Cheever . .$1 50 

The " heart of the Pacific, as it was and is," shows most vividly the 
contrast between the depth of degradation and barbarism, and the light 
and liberty of civilization, so rapidly realized in these islands under the 
humanizing influence of the Christian religion. Illustrated. 

The Republic of Liberia— Stockwell, . . . 1 25 

This volume treats of the geography, climate, soil, and productions 
of this interesting country en the coast of Africa, with a History of 
its early settlement. Our colored citizens especially, from whom the 
founders of the new State went forth, should read Mr. Stockwell's 
account of it. It is so arranged as to be available for a School Reader, 
and in colored schools is peculiarly appropriate as an instrument of 
education for the young. Liberia is likely to bear an important part 
in the future of their race. * 

Ancient Monasteries of the East— Curzon ■ 1 so 

_ The exploration of these ancient seats of learning has thrown much 
light upon the researches of the historian, the philologist, and the theo- 
logian, as well as the general student of antiquity. Illustrated. 

Discoveries in Babylon & Nineveh— Layard 1 75 

Valuable alike for the information imparted with regard to these most 
interesting ruins, and the pleasant adventures and observations of the 
aothor in regions that to most men seem like Fairyland. Illustrated. 

A Run Through Europe— Benedict, • • . . 2 00 

A work replete with instruction and interest. 

St. Petersburgh— Jermann 1 00 

Americans are less familiar with the history and social customs of the 
Russian people than those of any other modern civilized nation. Oppor- 
tunities such as this book affords are not, therefore, to be neglected. 

The Polar Regions— Osborn • 1 25 

A thrilling and intensely interesting narrative of one of the famous ex- 
peditions in search of Sir John Franklin — unsuccessful in its main object, 
but adding many facts to the repertoire of science. 

Thirteen Months in the Confederate Army 75 

The author, a northern man conscripted into the Confederate service, 
and rising from the ranks by soldierly conduct to positions of responsi- 
bility, had remarkable opportunities for the acquisition of facts respect- 
ing the conduct of the Southern armies, and the policy and deeds of their 
leaders. He participated in many engagements, and his book is one of 
the most exciting narratives of adventure ever published. Mr. Steven- 
son takes no eround as a partizan, but views the whole subject as with the 
eye of a neutral — only interested in suDserving the ends of history by the 
coutributiou of impartial facts. Illustrated. 

56 



J\fatio?ial School library. 



LIBRARY OF REFERENCE. 
Home Cyclopaedia of Literature & Fine Arts $3 oo 

A complete index to all terms employed in belles lettres, philosophy, theology, 
law, mythology, painting, music, sculpture, architecture, and all kindred arts. 

The Rhyming Dictionary— Walker . . . . 1 25 

A serviceable manual to composers, being a complete index of allowable rhymes. 

The Topical Lexicon— Williams i 75 

The useful terms of the English language classified by subjects and arranged ac- 
cording to their affinities of meaning, with etymologies, definitions and illustra- 
tions. A very entertaining and instructive work. 

Mathematical Dictionary— Davies & Peck • 5 oo 

A thorough compendium of the science, with illustrations and definitions. 



RELIGIOUS LIBRARY. 
The Service of Song— Stacy $1 50 

A treatise on Singing, in public and private devotion. Its history, office, and 
importance considered. 

True Success in Life— Palmer ...... $i so 

Earnest words for the young who are just about to meet the responsibilities and 
temptations of mature life. 

"Remember Me"— Palmer i so 

Preparation for the Holy Communion. 

Chrysostom, or the Mouth of Gold— Johnson X oo 

An entertaining dramatic sketch, by Rev. Edwin Johnson, illustrating the life 
and times of St. Chrysostom. 

The Memorial Pulpit— Robinson. 2 vols., each l 50 

A series of wide-awake sermons by the popular pastor of the Memorial Presby- 
terian Church, New York. 

Responsive Worship— Budington 60 

An argument in favor of alternate Scripture reading by Pastor and Congregation. 

Lady Willoughby l oo 

The diary of a wife and mother. An historical romance of the seventeenth cen- 
tury. At once beautiful and pathetic, entertaining and instructive. 

Favorite Hymns Restored— Gage hM 

Most of the standard hymns have undergone modification or abridgment by com- 
pilers, but this volume contains them exactly as written by the authors. 

Poets' Gift of Consolation • l so 

A beautiful selection of poems referring to the death of children. 

57 



National School Z,ibraty. 



VALUABLE LIBRARY BOOKS. 
The Political Manual— Mansfield fi 25 

Every American youth should be familiar with the principles of the government 
under which he lives, especially as the policy of this country will one day oall upon 
him to participate in it, at least to the extent of his ballot. 

American Institutions— De Tocqueville . . l 5Q 
Democracy in America— De Tocqueville . . 2 50 

The views of this distinguished foreigner on the genius of our political institu- 
tions are of unquestionable value, as proceeding from a standpoint whence we sel- 
dom have an opportunity to hear. 

Constitutions of the United States .... 2 25 

Contains the Constitution of the General Government, and of the several State 
Governments, the Declaration of Independence, and other important documents 
relating to American history. Indispensable as a work of reference. 

Public Economy of the United States ... 2 25 

A full discussion of the relations of the United States with other nations, espe- 
cially the feasibility of a free-trade policy. 

Grecian and Roman Mythology— Dwight • 3 oo 

The presentation, in a systematic form, of the Fables of Antiquity, affords most 
entertaining reading, and is valuable to all as an index to the mythological allusions 
so frequent in literature, as well as to students of the classics who would peruse in- 
telligently the classical authors. Illustrated. 

General View of the Fine Arts— Huntington l 75 

The preparation of this work was suggested by the interested inquiries of a 
group of young people concerning the productions and styles of the great masters 
of art, whose names only were familiar. This statement is sufficient index of its 
character. 

The Poets of Connecticut— Everest .... l 75 

With the biographical sketches, this volume forms a complete history of the 
poetical literature of the State. 

The Son of a Genius— Hofland W> 

A juvenile classic which never wears out, and finds many interested readers in 
every generation of youth. 

Sunny Hours of Childhood ? 5 

Interesting and moral stories for children. 

Morals for the Young— Willard 75 

A series of moral stories, by one of the most experienced of American educators. 
Illustrated. 

Improvement of the Mind— Isaac Watts • • 50 

A classical standard. No young person should grow up without having perused 
it 

58 



Church Music, etc. 



PUBLIC WORSHIP. 



Songs for the Sanctuary, 13 50 

By Ret. C. S. Robinson. 1344 Hymns, with Tunes. The most successful modern 
hymn and tune-book, for congregational singing. More than 200,000 copies have 
been sold. Separate editions for Presbyterian, Congregational, and Baptist 
Churches. Editions without Tunes, $1.75; in large type, $2.50. Abridged edition 
(" Songs for Christian Worship "), 859 Hymns, with Tunes, $1.50. Chapel edition 
607 Hymns, with Tunes, $1.40. 

International Singing Annual, 25 

Metrical Tune Book, l oo 

To be used with any hymn-book. By Philip Phillips. 

Baptist Praise Book, 2 so 

By Rev. Dbs. Fuller, Levy, Phelps, Fish, Armttage, Winkler, Etabts, Lor- 
cher and Manly, and J. P. Holbrook, Esq. 1311 Hymns, with Tunes. Edition 
without Tunes, $1.75. Chapel edition, 550 Hymns, with Tunes, $1.25. 

Plymouth Collection, 2 so 

(Congregational.) By Rev. Henry Ward Beecheb. 1374 Hymns, with Tunes. 
Separate edition for Baptist Churches. Editions without Tunes, $1.25 and $1.75. 

Hymns of the Church, 2 75 

(Undenominational.) By Rev. Drs. Thompson, Vermilye. and Eddy 1007 
Hymns, with Tunes. The use of this book is required in all congregations of the 
Reformed Church in America. Edition without Tunes, $1.75. Chapel edition 
("Hymns of Prayer and Praise "), 320 Hymns, with Tunes, 75 cts. 

Episcopal Common Praise, 2 ;s 

The Service set to appropriate Music, with Tunes for all the Hymns in the Book 
of Common Prayer. 

Hymnal, with Tunes, 1 25 

(Episcopal.) By Hall & Whiteley. The new Hymnal, set to Music. Edition 
with Chants, $1.50. Edition of Hymns only (" Companion " Hymnal), 60 cts. 

Quartet and Chorus Choir, 3 oo 

By J. P. Holbrook. Containing Music for the Unadapted Hymns in Songs for 
the Sanctuary. 

Christian MelodieS. ByGEO.B.CHEEVER. Hymns and Tunes. 1 00 

Mount Zion Collection. ByTEPERKiNs. Forthechoir. 1 25 

Selah. By Thos. Hastings. For the Choir. 1 25 



Public Worship (Partly Eesponsive) .... $1 00 

Containing complete services (not Episcopal) for five Sabbaths ; for use in schools, 
public institutions, summer resorts, churches without a settled pastor ; in short, 
wherever Christians desire to worship — no clergyman being present. 

The Union Prayer Book, 2 so 

A Manual for Public and Private Worship. With those features which are ob- 
jectionable to other denominations o f Christians than Episcopal eliminated or 
modified. Contains a Service for Sunday Schools and. Family Prayers. 

The Psalter, 16mo, 60 cts.; 8vo, 90 

Selections from the Psalms, for responsive reading. 

59 



School Furniture, 



FURNITUR 



(SUPPLIED BY THE NATIONAL SCHOOL FURNITURE CO.) 

PEARD'S PATENT FOLDING DESK AND SETTEE. 

This great improvement for the school-room has come already into such astonish- 
ing demar.d as to tax the utmost resources of the company's two factories to sup- 
ply it. By a simple movement the desk-lid is folded away over the hack of the 
settee attached in front, making a false back, and at once converting the school- 
room into a lecture or assembiy-room. When the seat also is folded, the whole 
occupies only ten inches of space, leaving room for gymnastic exercises, marching, 
etc., or for the janitor to clean the room effectively. 

NATIONAL STUDY DESK AND SETTEE. 

When not in use for writing, the desk-lid slides back vertically into a chamber, 
leaving in front an "easel," with clamps, upon which the student places his book 
and studies in an erect posture. As a folding-desk this offers many of the same 
advantages as the " Peard." 

THE GEM DESK AND SETTEE. 

Fixed top, and folding seat. This is the neatest pattern of the Standard School 
Desk, and the strongest in use. 

THE ECONOMIC DESK AND SETTEE. 

This is the cheapest good desk, with stationary lid and folding seat. 

All descriptions of 

HIGH SCHOOL DESKS, SCHOOL SETTEES, 

TEACHERS' DESKS, CHURCH SETTEES, 

BLACKBOARDS, PEW ENDS, 

CHAIRS, LECTERNS, Etc. 

Aleo, 

CLASS AND LECTURE CHAIR. 

The difficulty of reconciling furniture appropriate for the Lecture-room or 
Church with that convenient lor the Sunday-school is an old one. This article 
effectually remedies it. It consists simply of a plan by which chairs of a somewhat 
peculiar shape are connected with a coupling. The rows of chairs thus adjusted 
may at pleasure and with ease be spread out" straight in one line, forming pews or 
benches ; or they may be bent in an instant into a semi-circular form to accomo- 
date classes of any size to receive instruction from teachers seated in their midst. 



For further particulars, consult catalogues of the National School Furniture Co. 
and the Taylor Patent Chair Co., which may be obtained of A. S. Barnes <fc Co. 

60 



The Peabody Correspondence. 

New York, April 29, 1867. 
To the Board of Trustees of the Peabody Educational Fund : 

Gentlemen — Having been for many years intimately connected with the educa- 
tional interests of the South, we are desirous of expressing our appreciation of the 
noble charity which you represent. The Peabody Fund, to encourage and aid 
common schools in these waf-desolated States, cannot fail of accomplishing a great 
and good work, the beneficent results of which, as they will be exhibited in the 
future, not only of the stricken population of the South, but of the nation at large, 
seem almost incalculable. 

It is probable that the use of meritorious text-books will prove a most effective 
agency toward the thorough accomplishment of Mr. Peabody's benevolent design. 
As we publish many which are considered such, we have selected from our list 
some of the most valuable, and ask the privilege of placing them in your hands for 
gratuitous distribution in connection with the fund of which you have charge, 
among the teachers and in the schools of the destitute South. 

Observing that the training of teachers (through the agency of Normal Schools 
and otherwise) is to be a prominent feature of your undertaking, we offer you for 
this purpose 5,000 volumes of the " Teachers' Library," — a series of professional 
works designed for the efficient self- education of those who are in tneir turn to 
teach others — as follows : — 

500 Page's Theory and Practice of Teach- 250 Bates' Method of Teachers' Institutes 
ing. 250 De Tocqueville's American Instit'ns 

500 Welch's Manual of Object-Lessons. 250 Dwight's Higher Christian Educat'n. 
500 Davies' Outlines or Mathematical 250 History of Education. 

Science. 259 Mansfield on American Education. 

250 Holbrook's Normal Methods of 25) Mayhew on Universal Education. 

Teaching. 250 Northend's Teachers' Assistant. 

250 Wells on Graded Schools. 250 Northend's Teacher and Parent. 

250 Jewell on School Government. 250 Root on School Amusements. 

250 Fowle's Teachers' Institute. 250 Stone's Teachers' Examiner. 

In addition to these we also ask that you will accept 25,000 volumes of school ■ 
books for intermediate classes, embracing — 
5,000 The National Second Reader. 5,000 Beers' Penmanship. 

5.000 Davies' Written Arithmetic. 500 First Book of Science. 

5,000 Monteith's Second Book in Geog- 500 Jarvis' Physiology and Health. 

raphy. 500 Peck's Ganot's Natural Philosophy. 

3,000 Monteith's United States History. 500 Smith & Martin's Book-keeping. * 

Should your Board consent to undertake the distribution of these volumes, we 
shall hold ourselves in readiness to pack and ship the same in such quantities and 
to such points as you may designate. 

We further propose that, should you find it advisable to use a greater quantity of 
our publications in the prosecution of your plans, we will donate, for the benefit 
of this cause, twenty-Jive lier cent, of the'usual wholesale price of the books needed. 

Hoping that our request will r~.eet with your approval, and that we may have 
the pleasure of contributing in this way to wants with which we deeply sympa- 
thize!, we are, gentlemen, very respectfully yours, A. S. BARNES' & CO. 

Boston, May 7, 1867. 
Messrs. A. S. Barnes & Co., Publishers, New York: 

Gentlemen— Your communication of the 29th ult., addressed to the Trustees of 
the Peabody Education Fund, has been handed to me by our general agent, the 
Rev. Dr. Sears. I shall take the greatest pleasure in laying it before the board at 
their earliest meeting. I am unwilling, however, to postpone its acknowledgment 
so lon^, and hasten to assure you of the high value which I place upon your gift. 
Five thousand volumes of your " Teachers' Library," and twenty-five thousand 
volumes of " School-books for intermediate classes," make up a most munificent 
contribution to the cause of Southern education in which we are engaged. Dr. 
Sears is well acquainted with the books you have so generously offered us, and 
nmtes with me in the highest appreciation of the gift. You will be glad to know, 
too, that your letter reached us in season to be communicated to Mr. Peabody, be- 
fore he embarked for England on the 1st inst., and that he expressed the greatest 
gratification and gratitude on hearing what you had offered. 

Believe me, gentlemen, with the highest respect and regard, vour obliged and 
obedient servant, ROBT. C. WLNTHROP, Chairman. 

61 



The National Series of Stand a?°d Sc7iool-jBooks. 



GENERAL INDEX TO 

A. S. Barnes & Co.'s Descriptive Catalogue. 



PAGE 

Acoustics 28 

Aesthetics 36 

Algebra 17 

Analysis 10 

Anatomy 32 

Apparatus 47 

Arithmetic IT, 18, 21 

Astronomy 21, 32 

Belles Lettres 36 

Bible 7, 23 

Biography 54 

Book-keeping 27 

Botany 30 

Calculus 8, 17, 18 

Cards (for Wall) 7, 47 

Chairs 60 

Charts 7,8,10,22, 47 

Chemical Apparatus 30 

Chemistry 30, 34 

Church Music 59 

Civtl Government 39, 58 

Classics 44 

Composition 38 

Copy Books. 22 

Criticism 36 

Definers 8 

Desks 60 

Devotion 48 

Dialogues 38 

Dictation 8 

Dictionaries 9, 17, 50, 57 

Drawing 26 

Elocution 7, 38 

English Grammar 10 

English Literature 36 

Ethics 39 

Etymology 8, 9 

Examples in Arithmetic 18 

Familiar Science 28 

Erench 40 

Furniture 60 

Games 25 

Geography 12, 14 

Geology 28, 34 

Geometry 17, 18 

German 40, 42 

Government 39, 58 

Grammar 10, 17,40, 42, 44 

Greek 44 

History 23, 25, 53 

Intellectual Philosophy 39 

International Review 63 

Keys 



.10, 17, 34 Zoology. 
62 



Latin 44 

Lexicons 9,17,50, 57 

Library 49, 58 

Literature 1, 7, 36, 52 

Logic 17, 39 

Map Drawing 12, 26 

Maps 15,47 

Mathematics 17, 21 

Mechanics S8 

Mental Philosophy 39 

Morals 39, 58 

Music 48, 59 

Mythology 44, 58 

Natural History 32, 50, 55 

Natural Philosophy 28, 34 

Natural Science 27-35, 55 

Navigation 17 

Object Lessons 30, 49 

Optics 28 

Orthography 1, 8 

Penmanship 22 

Pens 22 

Philosophy, Intellectual 39 

Do. Natural 28, 34 

Physiology 32 

Poetry 36, 52 

Political Economy 36, 58 

Political Science 39, 58 

Prayer .- 48, 59 

Primers 1 

Readers 1-7 

Records 46 

Rhetoric 38 

School Library 52-58 

Settees 60 

Slated Books 17, 21 

Spanish 40 

Speakers 38 

Spellers 1,8, 9 

Surveying 17 

Synonyms 9 

Tablets 7, 47 

Tactics ... 46 

Teachers' Library 49-51 

Teachers' Monthly A 

Travel 56 

Trigonometry 17, 18 

Virgil 44 

Writing 8, 9. 29 

Written Spelling 1, 8, 9 

31 



